Liverpool, who have won the competition a record seven times, welcomed back skipper Steven Gerrard as a late substitute after six months out with a groin injury with a 2-1 away win at Championship side Brighton & Hove Albion.
Extra time was being played in the west London derby between Chelsea and Fulham which finished goalless after 90 minutes after Fulham's Pajtim Kasami missed a second half penalty, awarded after Alex, who was sent off, brought down Kerim Frei.
Hargreaves, making his debut for City, scored their opener after 17 minutes with a rasping shot from outside the penalty area, his first goal since April 2008 when he scored Manchester United's winner in a 2-1 victory over Arsenal.
Hargreaves last played a full 90 minutes of competitive soccer exactly three years ago on Sept. 21 2008, but has battled back to fitness after a series of career-threatening injuries.
Balotelli virtually wrapped up the victory with a low strike after 38 minutes, sending Birmingham, fielding just Keith Fahey from the side that won the trophy last season, out of the competition.
Liverpool endured a tough match on the south coast against Gus Poyet's championship high fliers, but an early goal from Craig Bellamy, and a second from Dirk Kuyt after 81 minutes put them in control. Ashley Barnes replied for Brighton with a 90th minute penalty.
In other matches Southampton beat Preston North End 2-1, Cardiff City and Leicester City were playing extra time after finishing level at 2-2 after 90 minutes while Everton and West Bromwich Albion was a late kickoff.
