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Goals in each half from Samir Nasri and Sergio Aguero put City in control, but Chelsea replied through Demba Ba, only for Roberto Mancini's men to resist the holders' desperate search for an equaliser.
Fifteen points behind Manchester United in the Premier League, City seem destined to relinquish their league crown, but they will be strong favourites to defeat Wigan Athletic in the FA Cup final on May 11 and reclaim the trophy they won in 2011.
For Chelsea, there will be no fifth FA Cup in seven seasons, and although there is the consolation of a Europa League semi-final with Basel, interim manager Rafael Benitez will hope his tired squad still have enough energy for the battle to win a Champions League place.
In stark contrast to Saturday's first semi-final between Wigan and Millwall, which took place beneath steady drizzle and in front of nearly 27,000 empty seats, Sunday's game was a sell-out played amid balmy spring temperatures.
City goalkeeper Costel Pantilimon was rewarded with a starting place for his performances earlier in the tournament, while Demba Ba returned to the Chelsea line-up after an ankle injury.
City started with intent and Chelsea goalkeeper Petr Cech had to produce a smart early save to deny Aguero when he prodded the ball goalwards after Gareth Barry's shot was helped on by Yaya Toure.
The Czech goalkeeper also repelled a near-post effort from Carlos Tevez, before Chelsea began to assert themselves.
Vincent Kompany came to City's rescue in the 23rd minute, heading the ball off the line after Eden Hazard's volley into the turf looped over Pantilimon, who had come for a cross and missed.
The Chelsea revival proved short-lived, however, and 10 minutes before half-time, City went ahead.
Toure slipped a pass inside to Aguero, who in turn found Nasri, and when the Frenchman's attempted return pass bounced back to him off Cesar Azpilicueta, he lashed the ball past Cech from close range.
James Milner then saw a goalbound shot hit Aguero and Kompany spurned a fine chance in first-half injury time, side-footing wide from 10 yards after Cech could only parry Milner's cross.
It was an opportunity that City were to rue for only two minutes of the second half, as Aguero doubled his side's advantage in the 47th minute.
After City worked a free-kick wide to the right, Barry swept an inswinging cross to the back post and Aguero directed a fine header back across goal and into the net via the right-hand upright.
Benitez gave his side a pair of fresh legs in the shape of Fernando Torres, still sporting a black mask to protect his broken nose.
Ba might have been at risk of being withdrawn, but instead it was John Mikel Obi who made way, and the Senegal striker vindicated Benitez's re-shuffle by reducing the arrears in the 66th minute.
It was a poor goal to concede, though, as David Luiz's long ball was allowed to bounce behind the City defence and Ba adjusted brilliantly to hook a half-volley past Pantilimon.
The Romanian goalkeeper then produced two important saves in quick succession, first swiping the ball off Juan Mata's toes and then thwarting Ba from close range after Hazard was allowed to manhandle Gael Clichy to the ground.
Luiz went fractionally close with a free-kick, while Torres felt he should have had a late penalty after going down under pressure from Kompany, but City survived.