Goals from tournament top-scorer Michael Drennan and Bradley Lewis were enough to guide the English club into the last eight after Djavan Anderson put the hosts ahead
The Dutch side went ahead early in the first half through Djavan Anderson, but strikes from Michael Drennan and Bradley Lewis secured victory in what was a superb comeback.
The visitors made the better start to proceedings, but it was Ajax who went ahead less than 15 minutes into the contest as Lesley de Sa pulled the ball back for Anderson to slot the ball home.
It was an advantage that lasted just over 20 minutes as Drennan levelled matters up with his sixth goal of the competition just minutes after seeing a curling effort travel just wide of the post.
Ajax were able to hold on until the break, but they eventually succumbed to the pressure from Villa midway through the second half as Lewis found the net to put the English side within touching distance of the last eight.
It proved to be the winner as the men from Amsterdam were unable to craft a response and the Villans held firm to keep their dream of reaching the final well and truly alive.