Swindon trailed 1-0 to a late first-half goal by Matt McClure, but scored twice the last 15 minutes to turn the tie on its head.
The hosts called the tune throughout the first half but poor finishing prevented them from taking the lead. Alex Pritchard spurned two good opportunities - one hitting the outside of the upright - and Jay McEveley headed wide.
They paid the penalty for their profligate finishing just before half-time when some calamitous defending allowed McClure to turn and fire Wycombe ahead with their only chance of the half.
It looked as through McClure's goal might proved a winner for the League Two visitors, but Swindon benefitted from a fortuitous 76th-minute equaliser.
Ryan Harley's 25-yard free-kick cannoned off Ajose, completely wrong-footing goalkeeper Matt Ingram and bouncing into the far corner.
With a penalty shoot-out looming, Swindon grabbed their winner.
McEveley's free-kick was headed back at the far post and Ajose spun on the turn to send Mark Cooper's men through.