Dagenham were the better side in the first half and deservedly led at the interval.
The only goal came with just five minutes on the clock as Scott Doe ran in at the near post to connect with a corner kick and head emphatically past Artur Krysiak in the Exeter goal.
Luke Wilkinson headed over the bar from another Dagenham corner and Medy Elito was off target with a shot from distance.
Exeter failed to create a single chance in a dismal first half. The only time Dagenham goalkeeper Jordan Seabright touched the ball was when he gathered a lame cross by Lawson D'Ath.
The second half was scrappy and stop-start with few chances for either side.
Dagenham frustrated Exeter with their time wasting, but when City did go close, Seabright somehow kept out Craig Woodman's superb free kick.
The win all but seals Dagenham's place in League Two for next season, while defeat for Exeter - their third in succession - leaves them looking nervously over their shoulders at their play-off rivals.