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 in a 2-1 away win at Championship (second division) high fliers Brighton & Hove Albion.
In the west London derby at Stamford Bridge 10-man Chelsea beat Fulham 4-3 on penalties after the sides drew 0-0 following extra time.
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.
Extra time was needed at Goodison Park where Everton came from behind to beat West Bromwich Albion 2-1 after Albion took the lead in the 57th minute with a Chris Brunt penalty.
Marouane Fellaini equalized 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 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.
Hargreaves last played a full 90 minutes of competitive soccer exactly three years ago on Sept. 21 2008. He has battled back to fitness after a series of career-threatening injuries and was given a standing ovation when he was replaced 12 minutes into the second half.
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.
As well as being a good night for Hargreaves, Kolo Toure also returned for the first time since completing a six-month ban for a failed doping test.
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.
Gerrard said he was delighted to be back, but told Sky Sports he would have come on before the 75th minute if Liverpool had been further ahead.
"If we were two or three up at halftime, I might have got a few more minutes, but it wasn't about me tonight, it was about progressing into the next round of the cup and getting (Sunday's 4-0 defeat) at Tottenham out of our system and I think we did that tonight."
