Liverpool entered the second leg with a 1-0 lead from the first match but came out attacking from the start, only to see City take the lead twice against the run of play.
Nigel de Jong put City ahead with a spectacular long-range strike in the 31st minute, but Liverpool captain Steven Gerrard equalised with a contentious penalty nine minutes later.
After Joe Hart made a string of saves to keep City in the game, the visitors again went in front through Edin Dzeko's easy tap-in from close range in the 67th minute.
But that lead was short-lived, as Bellamy's low shot from inside the area gave Liverpool the draw it needed to reach Wembley.