The visitors took a first-half lead through Tim Cole before Billy Jones pulled the Grecians level on the stroke of half-time from the penalty spot.
Lee Phillips put Exeter ahead just after the restart and Matthew Gill made sure in the final minute.
Dagenham goalkeeper Tony Roberts put in a man-of-the-match performance and made a string of fine saves throughout the game.
Roberts leapt well to keep out a long-range strike from Phillips in the opening minutes and also kept out Dean Moxey's close-range header.
Against the run of play it was the visitors who opened the scoring when Cole headed home a Sam Saunders corner.
In injury time Moxey was tripped inside the box by Anwar Uddin and Jones drilled the ball decisively past Roberts.
After the break Exeter took just over two minutes to take the lead when Jones whipped in a low cross which picked out Phillips who fired home from close range.
Dagenham chased the game and striker Craig Mackail-Smith forced Martin Rice to make a rare save and then fired wide from a tight angle.
In the last minute Phillips stole the ball from Cole and fed Gill who tapped home from 12 yards to make sure of the points.
Exeter boss Alex Inglethorpe said: "I was delighted with the performance. The response of the players to going a goal down was fantastic.
"I thought we played some good football and the goal from Lee Phillips was one of the best of the season."