Alex Nicholls gave Paul Tisdale’s side the lead eight minutes before half-time, neatly converting Tom Nichols’ cross.
Wheeler was among the goals in the Grecians’ 3-2 opening-day win over Yeovil and repeated those heroics with three minutes of the first half remaining.
The midfielder tapped home from close range after Nicholls had seen his initial effort saved by Swindon goalkeeper Lawrence Vigouroux following a mazy run.
Swindon named Nathan Byrne in their starting line-up despite interest from Championship clubs QPR and Brentford but he made little impact.
Jonathan Obika reduced the deficit with a stunning 25-yard shot in the 65th minute to set up a grandstand finish.
Jordan Williams almost took the game to extra-time when he fired against a post with 15 minutes remaining.
Swindon piled on the pressure in the closing stages but could not conjure an equaliser.