City where dealt an early blow with the news that Danny Sturridge, the first leg goalscorer, missed out after picking up a knee injury playing for the City senior side on Saturday.
It is 22 years since City last won this competition and when defender Ray McGivern inadvertently diverted the ball into his own net to hand Chelsea the advantage it looked like the wait would continue.
But shortly afterwards captain Ben Mee headed home Andrew Tutte's right-wing cross for a deserved leveller.
Vladimir Weiss was then brought down and the Slovakian winger gave City a half time lead with a superbly flighted free kick.
Citizen's boss Sven-Goran Eriksson and several members of the first team squad then had to endure a second half of constant Chelsea pressure as they chased an equaliser.
With three minutes remaining a rare City breakaway saw Weiss brought down by Chelsea's Spanish striker Sergio Tejera for a penalty.
Victory was confirmed as David Ball easily converted from 12 yards.