Christian Ribeiro scored the winning goal 11 minutes from time to consign Stanley to only their second away league defeat since mid-October.
The Reds took the lead through in-form Shay McCartan and also hit the woodwork twice but it was Exeter who made their chances count in a lively encounter played in tricky conditions.
John Coleman stuck with the same XI who overcame Hartlepool 3-1 midweek as they looked to overcome the Grecians, who lost to Liverpool on Wednesday in the FA Cup.
Tom Nichols and Jordan Moore-Taylor both tested Jason Mooney in the early stages but it was Stanley who scored in their first attack.
Double hero from midweek McCartan got the ball, raced ahead of the Exeter defence, rounded keeper Bobby Olejnik and clipped home the opener on three minutes.
Seamus Conneely tested Olejnik as the Reds looked for a second but Stanley's early lead only lasted two minutes as Tom Nichols latched onto a ball forward, and he fired past Mooney to level the scores.
The Grecians then enjoyed a spell of pressure with their best chance falling to ex-Stanley loanee Jayden Stockley with Mooney having to finger-tip his header over the bar.
At the other end, Sean McConville got the ball in the box and forced a save out of Olejnik while the in-rushing winger blazed over the bar after getting on the end of Piero Mingoia's cross.
Three minutes after the re-start, McCartan had a great chance to secure another double but his initial effort was blocked by Troy Brown and his second crashed against the woodwork and was cleared.
Stanley were enjoying the better of the exchanges with a Billy Kee header cleared for a corner while an Adam Buxton free kick caused chaos in the box but no one could capitalise.
At the other end, the Stanley defence had to block a Joel Grant effort while sub Arron Davies saw his strike crawl inches wide of the far post on 70 minutes.
The Reds were denied by the woodwork again when McConville's free kick found the head of an Exeter player and his flick hit the top of the crossbar before the ball was eventually cleared.
And the Stanley players had their heads in their hands on 79 minutes when, following a corner, Stockley rifled the ball goalwards with Mooney doing well to deny him, the ball bobbled around the box and it was sub Ribeiro who poked the ball home for the winner.
Coleman brought on Terry Gornell and Marcus Carver in the final 10 minutes but the Reds couldn't get the all-important equaliser.