With the game finely balanced with 15 minutes remaining, a poorly taken Aldershot free-kick was intercepted and from the resulting counter-attack Cureton fired home Liam Sercombe's parried effort.
Exeter - who brought in Tommy Doherty in place of Matt Oakley - took the lead in the 14th minute when Alan Gow slid in Cureton, whose powerful right-footed drive flew past Jamie Young.
But the Shots responded two minutes later when summer signing Craig Reid reacted quickest to bundle the ball over the line after an Anthony Tonkin free-kick was not cleared.
Exeter's Scot Bennett saw his header cannon back off the post after half an hour and Cureton should have done better when through on goal in stoppage time but was stopped by Sonny Bradley.
The second-half saw very few chances created but Cureton's well taken finish gave the Grecians their first win of the campaign.