The former Canaries boss, who is in dispute with his old employers after an acrimonious departure during the summer, enjoyed the last laugh as substitute Andreas Weimann hit a second-half brace to turn the tie around after Brett Holman had cancelled out Steve Morison's opener.
Christian Benteke cracked in a fourth during stoppage time to move Villa within 180 minutes of Wembley.
"I thought we were excellent for the whole game," Lambert said.
"We have been playing like that since the Sunderland game, but I am delighted, I really am."
Lambert had handed Darren Bent a rare start, only for the England striker, who saw an early one-on-one saved, to limp off just after the hour with a hamstring problem.
Lambert said: "I thought he looked pretty sharp. It was a blow for us, I don't know how serious it is."