Chelsea took a seventh-minute lead through an own goal by City left-back Ryan McGivern but City skipper Ben Mee and the impressive Vladimir Weiss hit back to give the Blues a deserved half-time lead.
David Ball's late penalty then sealed a 4-2 aggregate win for City, who were playing in their second final in three years after losing to
City striker Danny Sturridge failed to recover from the injury he suffered in the first-team's 2-1 win at Sunderland on Saturday so Robert Mak was the only change from the first leg.
And it was the visitors that claimed an early lead as Jacob Mellis cut between City defenders Dedryck Boyata and Mee, and although goalkeeper Greg Hartley was out quickly to make a block, the rebound hit McGivern and went in.
City replied with Ball, Mak and Andrew Tutte testing
The hosts remained on top with Donal McDermott firing over after a fine City move before Weiss was brought down by Chelsea captain Seth Nana Ofori-Twumasi in the 35th minute and the Slovakian winger stepped up to curl a sweetly-struck free-kick past Taylor.
City failed to maintain that fluency in a nervy second half, allowing Chelsea to come back into the game with Mellis and Miroslav Stoch both firing narrowly wide.
But as
Chelsea then showed their frustration in stoppage time as Mellis was sent off for a late challenge on Kieran Trippier. The City right-back was stretchered off but was fit enough to join the celebrations later, when Mellis apologised for his tackle.
Chelsea sub Adam Phillip also put in a crude challenge on McGivern as the clock ticked down but the young Blues admirably kept their composure.
Chelsea
Attendance: 19,753.