Both sides set out to attack and it was the visitors who enjoyed the best of the opening stages with Darren Bent and Fabian Delph going close.
However, it was the Canaries who took the lead with Steve Morison slotting home coolly after 19 minutes but it didn't take long for the Villains to get back in the game and Brett Holman equalised with a sumptuous half volley just two minutes later.
The match carried on in the same vein, with Morison coming closest to putting his side ahead but no-one could find the net and the sides headed in level at the break.
Much like the opening 45, both teams showed no fear in the opening stages of the second stanza and once again Morison was the biggest threat as he saw a goal ruled out for offside.
However, it was Villa who made the breakthrough as Andreas Weimann fired home from Eric Lichaj's cross after 79 minutes.
This understandably prompted Norwich to attack and their positivity cost them with Weimann again finding the net after good work from Benteke, before the Belgian rounded off the scoring following some wonderful build-up play.
Villa can now look forward to a semi-final tie and will hope the victory inspires them in their fight against relegation.