But both sides missed a string of chances before Exeter's resilient centre-back Santos Gaia headed home Alex Jeannin's corner.
The visitors doubled their lead when Gareth Sheldon finally beat Tony Roberts in the Dagenham goal from distance on his third attempt.
Exeter were cruising but the home side clawed one back just before half time through Craig Mackail-Smith.
Chris Moore got the home side level just after the break when he was pulled down in the box and got up to score the resulting penalty.
The game swung Dagenham's way just 10 minutes later when Exeter keeper Martin Rice was dismissed for bringing down Smith.
Moore stepped up again, but was denied by the substitute keeper, Paul Jones, who saved well down to his left.
Both sides had chances to win the game but it was the visitors who snatched a late winner through second-half substitute Wayne O'Sullivan.