An early Ross Barkley goal at the Etihad Stadium put Everton in sight of a Wembley date with rivals Liverpool but City replied through Fernandinho.
Controversy then erupted as Raheem Sterling then appeared to take the ball out of play before teeing up substitute Kevin De Bruyne to level the tie, but Sergio Aguero completed City's fightback.
The major downside for title-chasing City was the loss of the influential De Bruyne through injury late on, with the Belgian carried off after appearing to get his studs caught in the turf.
That took the shine off City's achievement but the team could still reflect on an excellent comeback after a compelling cup tie.
After their late push to sell tickets, City tried to stoke up the atmosphere with loud pyrotechnics while fans brought out their 1980s throwback inflatable bananas in their hundreds.
City controlled the majority of possession but Everton looked dangerous on the counter-attack.
It was from such a break the visitors took the lead after 18 minutes. As Aguero appealed for a foul after a tumbling under a Ramiro Funes Mori challenge, Everton quickly surged forward and fed the dangerous Barkley.
City had begun with Yaya Toure playing deep at the base of a midfield diamond, but the Ivorian was caught out of position as Barkley raced into space. Nicolas Otamendi was also outpaced as Barkley reached the edge of the area and fired a low shot past Willy Caballero, the City goalkeeping choice in cup football.
Trailing 3-1 on aggregate, City needed to respond and did so quickly. They looked to supply Aguero at every opportunity and David Silva found him with a clever flick. The Argentinian's long-range shot was blocked but Fernandinho followed up and fired home via a deflection off Leighton Baines.
Aguero rattled the post with a ferocious effort soon after and Joel Robles produced a brilliant reflex save as Silva pounced on Phil Jagielka's failed clearance.
The contest was a gripping and entertaining one and the pace was maintained after the break as Barkley broke clear and tested Caballero before Aguero got into position, only to miskick.
Silva became the second City player to hit the woodwork when he headed against the post from a Pablo Zabaleta cross and Toure could not quite reach the rebound.
Everton began to look like a team clinging on as Aguero almost turned in a cross from Sterling and the latter then had a shot blocked.
The goal City's play warranted, and levelled the tie, came after 70 minutes - but with a huge slice of controversy.
Sterling appeared to do well as he reached the byline to cross for substitute De Bruyne to rifle home - but the ball had gone out of play before it was pulled back by the England forward. None of the officials spotted it and Everton were left to rue their luck.
After being controversially denied a penalty in the first leg - while Sterling himself had a strong shout for a spot-kick turned down in the league match between the clubs a fortnight ago - City took full advantage.
Six minutes later Aguero put City ahead on aggregate with a superb glancing header from a pinpoint De Bruyne cross.
The tension was evident in the closing minutes as players from both sides flew into tackles but City held on, although the loss of De Bruyne was a late blow.
Manchester City's League Cup Final record
We take a look back at City's previous four finals in the competition.
1970 - Manchester City 2 West Brom 1
City won their second trophy in the space of nine days after beating Poland's Gornik Zabrze to lift the European Cup Winners' Cup. The 2-1 scoreline was repeated after Jeff Astle had opened the scoring for Albion, Mike Doyle sending the game into extra time before Glyn Pardoe grabbed the winner.
1974 - Wolves 2 Manchester City 1
City suffered late heartbreak four years later after Kenny Hibbitt had given Wolves the lead on the stroke of half-time. Colin Bell equalised just before the hour-mark and the game appeared destined for extra time until John Richards struck the winner with five minutes remaining.
1976 - Manchester City 2 Newcastle 1
Dennis Tueart wrote his name into City folklore with his Wembley winner, a stunning overhead kick. Teenage winger Peter Barnes had given City an early lead before Alan Gowling levelled, but Tueart stole the headlines straight after half-time with a spectacular effort which saw Tony Book became the first man to win the competition as player and manager.
2014 - Manchester City 3 Sunderland 1
City marked their 38-year absence from the League Cup final with a comeback victory over Sunderland. Fabio Borini fired the Black Cats into a 10th-minute lead but two goals in 60 seconds after the break, from Yaya Toure and Samir Nasri, turned the tide City's way before Jesus Navas settled the matter with a stoppage time counter-attack.
Manchester City boss Manuel Pellegrini is confident his side can turn around their Capital One Cup semi-final against Everton.
City host the Toffees in the second leg of their semi-final on Wednesday needing to overcome a 2-1 deficit to claim a place in next month's Wembley final.
It is the third meeting between the two clubs in three weeks, after they also played out a goalless draw in the Barclays Premier League, and Pellegrini is hopeful of getting the upper hand this time.
The Chilean said: "I am sure if we repeat what we did against Everton here in the last game in the Premier League, especially the last 45 minutes, we are going to score goals.
"But it's very important to concentrate on defending because they have very dangerous attacking players - Ross Barkley, (Romelu) Lukaku, (Gerard) Deulofeu - difficult players to stop.
"We must be a very balanced team but not change our style of what is the best way to win."
The match at the recently-expanded 55,000-capacity Etihad Stadium is not yet a sell-out but, with 48,000 tickets sold as of Tuesday, the attendance will still be City's largest for a domestic cup tie since 1981. If the crowd goes above 50,000, as the club hope, it will be City's biggest cup attendance for 41 years.
Pellegrini said: "I'm sure that we're going to have the support of the fans we have in every game here.
"Of course it's asking a lot of supporters to support every game, but normally here in England the stadiums are always full for different reasons."
Everton go into the game with an advantage and the extra incentive of a possible all-Merseyside final against Liverpool to aim for.
Manager Roberto Martinez is not fazed by City or their liking for playing at home. City have failed to score just once in their last 26 games at Eastlands - and that was against Everton a fortnight ago.
Martinez said: "City have one of the best home records in the league and we know it is a massive hurdle but we will give everything we have got to get to the final.
"Our away record is exceptional - one defeat all season.
"This is a one-off and one that we have to be prepared. Mentally it is a huge occasion.
"We need to make sure we play the game to win that game in isolation, you can only think about the overall score in the last five or 10 minutes because there is not a big enough margin to be thinking about it."
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