There was a moment of doubt over the decisive penalty from Fulham's Bryan Ruiz whose shot hit the bar, bounced on the line and hit the bar again before bouncing away. Fulham protested the ball had crossed the line, but the officials ruled otherwise.
Fulham had a great chance to seal what would have been their first win at Chelsea since 1979 but Pajtim Kasami missed a second half penalty, awarded after Alex, who was sent off, brought down Kerim Frei.
Marouane Fellaini equalised for Everton in the 89th minute with skipper Phil Neville smashing in Everton's winner after 103 minutes, only his fifth goal in 247 appearances for the Toffees.
In the two other matches, Championship leaders Southampton beat in-form Preston North End of League One 2-1 while Cardiff City beat Sven-Goran Eriksson's Leicester City 7-6 on penalties after the all-Championship tie ended 2-2 following extra-time.
Hargreaves scored Man City's 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.
Balotelli effectively wrapped up the victory with a low strike after 38 minutes, sending Birmingham, fielding just Keith Fahey from the side that beat Arsenal in last season's final, out of the competition.
Liverpool endured a tough match on the south coast against Gus Poyet's championship hopefuls, but an early goal from Craig Bellamy, his first since his return to Liverpool, and a second from Dirk Kuyt after 81 minutes put them in control. Ashley Barnes replied for Brighton with a 90th-minute penalty.
"If we were two or three up at half-time, I might have got a few more minutes, but it wasn't about me tonight, it was about progressing int
