The game marked the first return to Carrow Road for Lambert, who was given a mixed reception from the home crowd, despite his consecutive promotions at the East Anglian club.
Villa, dumped out at this stage two years ago, were looking for a distraction from Premier League troubles against the Canaries, who had reached the last eight for the first time since 1995/6.
Norwich came into the game on a fantastic run of form, unbeaten in ten games in all competitions and sat just four points off the top four, whilst Villa had won one of their last sixteen away games against Premier League opposition and scored just two goals in their last five games.
The game started brightly, and Norwich could have been ahead after less than two minutes, a cross from Grant Holt hitting the far post after looping over Shay Given. Moments later Darren Bent was sent through on goal by Brett Holman but fired straight at Mark Bunn. Australian Holman was comfortably the best player during the first half, his range of passing superior to any other performer.
Given, back in the side after losing his place for Premier League football to Brad Guzan, looked out of confidence as his kicking occasionally got Villa into trouble, and the former Irish international was beaten on 19 minutes when Steve Morison collected a long ball from Holt and comfortably knocked the ball into the goal, Given left stranded.
Incredibly, it was only 75 seconds until Villa equalised, Holman collecting a knock down from Christian Benteke before half-volleying superbly into the bottom corner.
The goal initiated a spell of pressure from the away side, and Mark Bunn was forced into a tremendous triple save from Holman, Benteke and Bent, the second save from the Belgian being the most impressive.
As the game settled before the break, the final action of note was the unfortunate withdrawal of Bent with what seemed to be a pulled hamstring. The striker had looked impressive and will feel frustrated that his chance to impress Lambert was cut short.
The second half began with a pair of disallowed goals as both Morison and Benteke had efforts ruled out for offside. Given then made a reaction save from Holt as both sides pushed for the lead.
It eventually came through substitute Weimann for Villa, who converted Eric Lichaj's cross from close range after 78 minutes. The Norwich defending was static as Weimann ghosted in, and seven minutes later the Austrian performed the trick once more, this time taking a pass from Benteke and slotting home. Villa had been rewarded for their invention and creativity during the second half, with both Weimann and Benteke impressing hugely.
One minute into stoppage time and Benteke scored the goal of the game, the away side putting together 17 passes before working the ball to the edge of the box, where the Belgian forward fired low into the right hand corner of the net.