Chelsea's attackers were too much for Manchester United's young defence as the hosts booked a match against Leeds in the Capital One Cup last eight.
Chelsea made two defensive errors as Ryan Giggs and Javier Hernandez scored either side of David Luiz's penalty.
Gary Cahill made it 2-2 but it appeared Nani's chipped finish would win it, before Eden Hazard's spot-kick.
Daniel Sturridge and Ramires both registered in extra time before Giggs added a late consolation penalty.
Luiz also hit the bar in the added half hour as Chelsea finished on top and were rewarded for introducing the cream of their talent from the bench
.