Murray, 26, raised a standing ovation on Centre Court where crowds roared their support at his straight sets 6-4 7-5 6-4 victory.
He broke down in tears of joy as his triumph sank in, before sprinting across the famous lawn to embrace his girlfriend in the player's box.
Murray, who held his trophy aloft and kissed it, said: "It feels slightly different to last year, last year was one of the toughest moments of my career.
"It was an unbelievably tough match, so many long games and I don't know how I managed to come through that.
"That final game was unbelievable, three match points."
Paying tribute to the huge support he has received, Murray said: "I understand how much everyone else wanted to see a British winner at Wimbledon so I hope you guys enjoyed it, I did my best."
Asked if he had forgotten mother Judy in his immediate celebrations, he admitted: "I did forget her, I just heard squealing behind me when I was trying to get down and I turned back."
Murray dedicated the game to coach Ivan Lendl, as well as praising his team.
He said: "I have got a great team, they have stuck by me through a lot of tough moments and this one is especially for Ivan as well.
"He did everything to try and win this when he was playing and I am glad I was able to help him out when he is coaching.
"He is a fantastic person, has worked very hard with me, is very patient with me because I am not easy at times."
Praising Djokovic, he said: "I played Novak many times and I think when everyone has finished playing he is going to go down as one of the biggest fighters.
"He has come back so many times from losing positions and he almost did it again today so that made it extra tough."
Murray was congratulated privately by the Queen following his historic win.
A Buckingham Palace spokeswoman said: "I can confirm that the Queen has sent a private message to Andy Murray following his Wimbledon victory."
The Prime Minister was in the Royal Box to watch Murray's straight sets victory.
As Centre Court erupted in joy, Mr Cameron tweeted: "It was a privilege to watch @andy-murray making history at Wimbledon, and making Britain proud."
Stars from the worlds of showbiz and sport also took to Twitter to hail Murray's achievement on a sweltering day in south west London.
Star Wars actor and fellow Scot Ewan McGregor wrote: "Oh my God. What a match, what a player. Inspiring a Nation. Andy Murray!! Ya Beauty."
Former England striker and Match Of The Day presenter Gary Lineker added: "I sense @andy-murray will take a bit of beating at Sport's Personality Of The Year! #wimbledon."
Murray's mother said she was an "emotional wreck" following his win but added: "I feel very proud of him."
She said: "That has just got to be the best feeling ever, to watch your son win Wimbledon."
Mrs Murray, who said she had been "shaking" for the past three hours, said Murray would now be feeling a mixture of "disbelief and relief and sheer joy".
"This is the thing he has talked about winning since he was a little boy playing tennis and of course it was a dream that has suddenly become a reality for him," she said.
"I think he will be absolutely over the moon."
She said she had a "good feeling" ahead of the match.
"He has been strong and composed all week, but Novak is such a tough competitor. To bury him you have to play unbelievably well," she said.
"I knew he (Murray) was in a good place and that if he played his best he had a chance to win but you are playing the best player in the world so you've just got to bring your A game and he brought a pretty decent game today."
She added: "They had been out there for three hours and it was so hot, but he was so determined to win it."
Murray's great friend Ross Hutchins, who was diagnosed with cancer last year, appeared stunned by the result and almost lost for words.
"I'm ecstatic," he said. "I am so pleased for him."
Andy Murray in his own words
AFTER WINNING THE US OPEN BOYS' TITLE IN 2004: "I've been compared to Tim (Henman) a lot and to be compared to someone who has been in the top 10 for six or seven years is great. I think I have got a chance of going all the way in tennis and that's what I want to do."
THINKING BIG IN 2005: "I want to get into the top 10 - then maybe I can get a date with Maria Sharapova!"
ON THE BRITISH MEDIA FOCUS, SPEAKING IN OCTOBER 2005: "I have watched Tim Henman go through it in the last 10 years, and I hope they [the media] are a bit nicer to me than they were to him."
AFTER LOSING IN THE 2006 AUSTRALIAN OPEN FIRST ROUND: "If you guys expect me to play well every single match and every single tournament then it's not going to happen."
AFTER SUCCEEDING TIM HENMAN AS BRITISH NUMBER ONE IN FEBRUARY 2006: "If you're number one in your country it's obviously nice but it doesn't mean anything to the other players, whereas if you're in the top 10 in the world that's pretty special."
ON RECEIVING A GOOD-LUCK CALL FROM SIR SEAN CONNERY DURING WIMBLEDON 2006: "There's nothing cooler than being woken up by James Bond."
ON WHO HE WOULD BE SUPPORTING AT THE 2006 WORLD CUP: "Anyone but England."
AFTER A BACKLASH TO HIS "ANYONE BUT ENGLAND" COMMENT: "I made a joke, I don't mind whether England win or lose!"
ON BROTHER JAMIE (speaking in 2005): "My brother is very talented. He was the number two junior in the world when he was 13, then he went down to an LTA school in Cambridge and they ruined him. It was their fault."
AFTER WINNING SAN JOSE TITLE IN 2006, HIS FIRST ON THE ATP TOUR: "It's the biggest moment in my life in tennis so far."
AFTER LOSING TO ROGER FEDERER IN THE 2008 US OPEN FINAL: "I'm disappointed right now. I could have done a few more things and given myself a few more opportunities."
AFTER LOSING TO FEDERER IN THE 2010 AUSTRALIAN OPEN FINAL: "I'm getting closer (to a grand slam). My results in the grand slams would show that. I've just got to keep working hard."
AFTER DEFEAT TO RAFAEL NADAL IN THE 2010 WIMBLEDON SEMI-FINAL: "I don't feel like I played terribly. I've had some good wins against Rafa where I played great tennis. But he played great, and that was the difference. There's a great player in the final."
AFTER LOSING TO NOVAK DJOKOVIC IN THE 2011 AUSTRALIAN OPEN FINAL: "He played great although I would have liked to have played better."
ON WORKING WITH NEW COACH IVAN LENDL AT THE 2012 FRENCH OPEN: "There's not been one radical change. A lot of it is minor details. But if you pick 10 small things to work on and change, that can turn into a big difference."
AFTER LOSING TO FEDERER IN THE 2012 WIMBLEDON FINAL: "I'm getting closer. If I wasn't emotional I guess I'm probably in the wrong sport."
FOLLOWING AN OLYMPIC FINAL WIN OVER FEDERER: "I've had a lot of questions asked about me many times. I'm just glad I managed to put on a performance I've been waiting for. To win, in the way I did, makes those losses a little bit easier to take."
AFTER DEFEATING DJOKOVIC TO WIN THE US OPEN, HIS MAIDEN GRAND SLAM, IN 2012: "It's great to have finally done it. I hope now it inspires some kids to play tennis and also takes away the notion that British tennis players choke or don't win or it's not a good sport."
AFTER LOSING TO DJOKOVIC IN THE 2013 AUSTRALIAN OPEN FINAL: "I know no one's ever won a slam, the immediate one after winning their first one. It's not the easiest thing to do. And I got extremely close."
AFTER WINNING WIMBLEDON FOR THE FIRST TIME IN 2013: "That win was for myself but I also understand how much everyone else wanted to see a British winner at Wimbledon. I hope you guys enjoyed it."
A man with iron will - timeline of Andy Murray's career
1987: Born May 15, Dunblane, Scotland.
1999: December - Wins junior Orange Bowl title in Miami.
2004: September - Becomes first British winner of US Open boys' title, beating Ukrainian Sergiy Stakhovsky.
2005: March - Becomes youngest British Davis Cup player, aged 17, in match against Israel.
June - Defeats Radek Stepanek to become first Scot to reach third round at Wimbledon. Loses to David Nalbandian.
October - Reaches first ATP final at Thailand Open, losing to Roger Federer.
2006: February - Claims first ATP title by beating Lleyton Hewitt in San Jose.
February - Moves above Tim Henman and Greg Rusedski in rankings to become British number one.
April - Splits from coach Mark Petchey.
July - American Brad Gilbert appointed Murray's new coach.
2007: May - Snaps a tendon in his wrist at the Hamburg Masters, and the injury forces him to miss the French Open and Wimbledon.
November - Splits from coach Gilbert.
2008: July - Loses to Rafael Nadal in Wimbledon quarter-finals.
August - Defeats Novak Djokovic in Cincinnati to claim first Masters Series title.
September - Beats top seed Nadal in semi-finals of US Open to reach first grand slam final, where he loses 6-2 7-5 6-2 to defending champion Federer.
2009: June - Loses to Fernando Gonzalez in French Open quarter-finals.
June - Becomes first British player since 1938 to win Queen's Club title, beating James Blake in the final.
July - Reaches first Wimbledon semi-final but loses 6-4 4-6 7-6 7-6 to Andy Roddick.
August - Wins Montreal Masters with victory over Juan Martin Del Potro, and becomes world number two for first time.
2010: January - Loses 6-3 6-4 7-6 to Federer in Australian Open final.
July - Loses Wimbledon semi-final 6-4 7-6 6-4 to Nadal. Splits with coach Miles Maclagan after two and a half years.
2011: January - Beaten 6-4 6-2 6-3 by Djokovic in second successive Australian Open final.
June - Loses 6-4 7-5 6-4 to eventual champion Nadal in French Open semi-final.
September - Loses 6-4 6-2 3-6 6-2 to Nadal in US Open semi-final.
December - Hires eight-time grand slam winner Ivan Lendl as his new coach.
2012: January - Loses 6-3 3-6 6-7 6-1 7-5 to Djokovic in Australian Open semi-final lasting four hours and 50 minutes.
June - Murray's French Open run ends in the quarter-finals with defeat by David Ferrer.
July - Reaches Wimbledon final for the first time with victory over Jo-Wilfried Tsonga, but loses 4-6 7-5 6-3 6-4 to Federer who lands a record-equalling seventh title. Murray breaks down in tears in his on-court interview, saying: "I'm getting closer."
August - Wins Olympic gold in singles at Wimbledon by beating Federer 6-2 6-1 6-4, and takes silver in mixed doubles with Laura Robson.
September - Finally breaks his grand slam duck, beating Djokovic 7-6 (12/10) 7-5 2-6 3-6 6-2 to win the US Open.
December - Finishes third in the BBC's Sports Personality of the Year awards.
Awarded an OBE in the New Year Honours list.
2013: January - After beating Federer in the semi-finals, Murray loses 6-7 (2/7) 7-6 (7/3) 6-3 6-2 to Djokovic in the Australian Open final.
May - Forced to pull out of the French Open due to a back injury.
June - Wins his third AEGON Championships at Queen's Club, beating Marin Cilic in the final.
July 5 - Reaches his second Wimbledon final, beating Poland's Jerzy Janowicz 6-7 (2/7) 6-4 6-4 6-3 in the semi-finals to set up a match against world number one Djokovic.
July 7 - Triumphs in the Wimbledon men's singles with a 6-4 7-5 6-4 victory over Djokovic, becoming Britain's first winner of the title since 1936.
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