The Hatters had looked like they were on the way to a much-needed three points after six straight defeats, only for the Grecians to net three times in the second period, with Wheeler popping up at the death.
Luke Guttridge had put Luton in front in the 21st minute, taking Mark Cullen’s pass to beat James Hamon.
Luton had a second when they broke away again on the stroke of half time, with Cullen taking Jonathan Smith’s pass to fire low into the net.
However Exeter were back in the game in the 50th minute when Ryan Harley produced a lovely volley into the roof of the net.
The Grecians had a deserved leveller with 14 minutes to go when centre-back Jordan Moore-Taylor slotted into the bottom corner when a free-kick fell invitingly into his path.
Paul Tisdale’s side then won it in the dying seconds, as Christian Ribeiro’s header hit the post, but Wheeler slid the ball in from close range.