The win was marred by a horrific injury to Ben Mulley in the 11th minute which saw Ayoub Enajar given a straight red card.
Sam Merson netted a 98th-minute penalty to keep Villa among the promotion play-off spots in fourth place.
A 0-0 draw means Villa will need six points from three remaining matches to confirm a promotion play-off berth.
Villa's three-match winning run was halted by goals from Leatherhead's Matthew Everitt and Kareem Akinnibi.
Villa are up to third and within touching distance of the promotion play-off spots.
It was 0-0 during the 26th minute when the incident happened.