Cureton notched a goal in each half as the Grecians bagged all three points after surviving a late scare as the bottom-of-the-table Bees threatened to bag a share of the spoils.
Jake Hyde was on the scoresheet for the home side in the 81st minute but it was not enough as the London club slumped to a seventh defeat in their last eight league outings.
Cureton opened the scoring 10 minutes before half-time when a flick-on from former Bees frontman John O'Flynn left him with time and space to pick his spot from close range.
And it was 2-0 after 67 minutes when Tommy Doherty picked out Cureton in the box - after more good work from O'Flynn - and the former Bristol Rovers striker made no mistake.
Hyde - fresh from notching against Fleetwood on Saturday - bagged a consolation nine minutes from time as the Bees threatened a grandstand finish but ultimately Exeter held on to bounce back from two consecutive defeats.