Scott Honeyball and Oliver Squires were cup-tied, Gary Phillips had to play despite suffering from flu, and Cliff Hercules returned from a lengthy lay-off as a substitute.
Steve Heard had given United a 33rd minute lead from a Phil Mason cross and Aylesbury should have added more in a quiet first half with City on the defensive throughout most of the half.
In the second half goalkeeper Paul Tassell made an incredible save from Heard, whilst Davy hit the bar at the other end. City's Danny Wise levelled on 74 minutes with an angled shot past Phillips.
In the closing stages Gary Crawshaw and Neil Selby both found the net, but both were ruled out thanks to the linesman's offside flag and the game went into extra time.
Adam Clarry and Crawshaw went close, but the tie was eventually settled after 109 minutes when Jason Davy sprinted clear and rounded Phillips before slotting the ball into the empty net.