The Sky Blues looked set to claim the points after striker Leon Clarke opened the scoring against his old club six minutes into the second half.
Yaser Kasim equalised for the hosts 13 minutes from time before substitute Storey grabbed a dramatic 86th-minute winner for Mark Cooper's promotion hopefuls.
Clarke famously quit Swindon after a tunnel bust-up with former manager Paolo Di Canio and came back to haunt the Robins by heading in Carl Baker's cross from close range.
Kasim levelled with an excellent strike from the edge of the penalty area before John Fleck rattled the crossbar for the visitors.
Storey claimed the victory with a delicious chipped shot from mid-way inside Coventry's 18-yard box.
The win could have been more emphatic had Kasim's deflected effort in the dying minutes not struck a post.