Striker Corr climbed off the substitutes' bench 14 minutes from time to galvanise a Swindon side that had fallen behind to two second-half goals.
Leicester captain Matt Oakley opened the scoring with a fine solo effort in the 57th minute and top-scorer Matty Fryatt slotted home his 15th goal of the season to make it 2-0 to the Foxes.
But Simon Cox grabbed a goal back with his 15th strike of the campaign after being set up by Lee Peacock in the 84th minute.
Corr then volleyed the ball into the roof of the net two minutes later after the Leicester defence had failed to clear a corner.
Leicester were left to rue a string of missed chances. Oakley twice went close from long range, while Fryatt blazed over the crossbar when well placed early on.