Murray played an instrumental role in helping Great Britain to secure a historic Davis Cup title in November.
Rugby league star Kevin Sinfield was the runner-up while world champion heptathlete Jessica Ennis-Hill came third.
Murray said: "Thank you very much, I genuinely didn't prepare anything, I didn't expect this.
"A friend actually sent me a message the other day with an article from a newspaper which said that 'Andy Murray is duller than a weekend in Worthing', which I thought was a bit harsh - to Worthing."
Dressed in a navy suit that matched the rest of his Davis Cup colleagues, Murray explained his wife Kim was unable to attend on account of her pregnancy.
He added: "It's very humbling to be up here in front of so many great athletes, I'm a huge sports fan and I'm extremely nervous right now, so apologies about that, but congratulations to everyone who was involved tonight.
"I would really like to thank my team, all my team-mates, all of the staff that were over there, they were absolutely incredible.
"This has been a five-year journey, we were down at the bottom level of tennis and now we're ranked number one in the world - I never thought that would be possible.
"I'd also like to thank my family, some of them are here, not all of them are here - my wife is at home, she's expecting a baby in a few weeks and is unable to travel, so I've got that to look forward to.
"I'd like to thank everyone who voted for me, I really appreciate that.
"I dedicate my life to this sport, I work as hard as I can every single day to try and make you proud and I appreciate all the votes - thank you very much."
Michael O’Neill wins Coach of the Year
Northern Ireland manager Michael O’Neill was named Coach of the Year.
For O’Neill, now one of the most respected coaches in the game, it was the perfect way to end what has been a dream 2015.
In October he guided Northern Ireland to the European Championship finals for the first time in the country’s history and the praise has been flowing in his direction ever since.
He was overwhelmed by the incredible reception he received from 7,500 people, all of whom stood to recognise his achievements as Portrush golfing hero Graeme McDowell and legendary athlete Denise Lewis presented him with his award.
Belfast coach Damien Lindsay lifts Unsung Hero award
West Belfast coach Damien Lindsay won the Get Inspired Unsung Hero award.
The Unsung Hero award honours individuals from the world of grassroots sport and recognises those who, on a voluntary basis, go the extra mile to contribute to sport in their local community. Damien formed St James Swifts FC to encourage local children to become involved insport as opposed to anti-social behaviour. He was selected from 15 regional winners from across the UK and was presented the award in his hometown by Eddie Izzard.
Tony McCoy collects lifetime achievement gong
Jockey AP McCoy received the Lifetime Achievement Award after bringing his illustrious career to a close in April.
McCoy - who won the main Sports Personality prize in 2010 - retired after riding 4,357 winners and being crowned Champion Jockey 20 consecutive times.
After receiving the award from fellow Northern Irishman Rory McIlroy, McCoy said: "Could I just say what an honour and a privilege it is to receive such a prestigious award, especially here in Belfast.
"Being from Northern Ireland, a place that has produced so many talented and successful sports people.. it makes tonight even more special.
"Racing is a wonderful sport. I was lucky to live my life as a jockey through it and for that I will be for ever grateful."
The decision to award the Moneyglass man the prestigious prize had been announced last week but that did not take away from the emotion and joy of the Northern Ireland public inside the SSE Arena seeing one of their own join an illustrious list of iconic sporting figures.
There was a prolonged standing ovation from the 7,000 crowd for the hugely popular 41-year-old jockey who retired earlier this year.
McIlroy, said: “It is very special. I look at the names on this award and I’ve proud to have my own name beside such global superstars. So many other people have had a hand in my sucesss. The owners, the trainers and the horses I was lucky enough to ride. I couldn’t have picked a better sport.”
Rugby star Dan Carter was named as Overseas Sports Personality of the Year before the event.
The Racing 92 fly-half, who helped New Zealand win the Rugby World Cup, staved off competition from big names like Usain Bolt, Novak Djokovic, Katie Ledecky, Jordan Spieth and Serena Williams after the award was decided by a public vote for the first time.
This year's Get Inspired Unsung Hero prize went to West Belfast man Damien Lindsay, who formed St James Swifts football club to prevent young people in his local area from being dragged into anti social behaviour like rioting.
Mr Lindsay was selected from 15 regional winners and was presented with the award by comedian Eddie Izzard.
Ellie Downie has become the second consecutive gymnast to carry off the BBC Young Sports Personality of the Year award.
The Nottingham 16-year-old succeeds Great Britain team-mate Claudia Fragapane, who won the award in 2014.
Downie became the first British female gymnast to win a major all-around medal when she won bronze at the European Championships.
And she followed that up by clinching team gold at the World Championships in Glasgow alongside elder sister Becky, Fragapane, Ruby Harrold and Amy Tinkler.
Downie joins an illustrious list of previous winners of the award, including Wayne Rooney and Tom Daley.
Bailey Matthews was the recipient of the Helen Rollason Award, which is presented for an "outstanding achievement in the face of adversity".
The eight-year-old from Yorkshire, who has cerebral palsy, completed a children's triathlon at Castle Howard in June.
Bookmakers have reported "one-way traffic" for Andy Murray to win the 2015 BBC Sports Personality of the Year title.
According to Coral, the tennis champion has odds of 8-13 while Jessica Ennis-Hill goes into the evening just behind, on 5-2.
Tyson Fury can be backed at 18-1.
Coral's John Hill said: "Over 70% of the bets we are taking today on tonight's award ceremony are for Andy Murray, who is the warm favourite to win this award after he helped Great Britain to their first Davis Cup success since 1936."
Twelve of the biggest names in sport will go head to head for the coveted award, which will be revealed during a live show in Belfast tonight.
Some 7,500 people are expected to pack into the SSE Arena for the star-studded ceremony.
Murray, 28, helped make history by guiding Great Britain to their first Davis Cup victory for almost 80 years, and heptathlete Ennis-Hill, 29, won her second world title in Beijing in August, just months after giving birth.
Others in the running include "king of the long jump" Greg Rutherford, who took gold at the World Championships, Formula One champion Lewis Hamilton, world road race cycling champion Lizzie Armitstead and runner Mo Farah, who also enjoyed success in Beijing, winning his third successive 5,000m world gold medal.
However, the inclusion on the shortlist of world heavyweight champion boxer Tyson Fury has sparked a major outcry.
Controversial comments he made allegedly equating homosexuality with paedophilia, joking about being violent to women, and saying a woman's "best place is on her back" have prompted gay and equal rights campaigners to mount protests outside the SSE Arena ahead of the main event.
It is the first time the event has been staged in Northern Ireland.
The awards are being hosted by Gary Lineker, Clare Balding and Gabby Logan.
Last year the ceremony took place in Glasgow, when Hamilton took the crown, with golfer Rory McIlroy in second place and athlete Jo Pavey in third.
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