The young Reds provided a stern test for cup holders Villa, who booked a quarter-final against Crystal Palace with their 1-0 triumph.
Wrexham fell behind in 12th minute as Bradley Burrowes broke the deadlock.
Villa looked to extend their advantage soon after via Max Jenner, but the determined young Reds managed to clear the danger.
Wrexham posed a threat as half time neared and they forged an opportunity a minute prior to the break.
Joe Rees played a free kick into the box and George Cruise’s effort was saved by Owen Asemota.
Chances were few and far between in the second half. For Wrexham, Sammy Chesworth played in Wes Graham, whose shot was deflected behind for a corner.
The Reds’ cup run started with a 5-1 home victory against Port Vale in the third round.
This was followed by their 3-1 success away to Southend United in the last-32, when Jac Jones, Carrick Murphy and Chesworth got the goals.