A Craig McAllister brace cancelled out Ben Strevens' effort as the Grecians recorded their first win of the year, whilst inflicting a first defeat in seven league games on the Daggers.
Dagenham began well, but couldn't find the killer blow as both Strevens and Matt Ritchie went close when well-placed.
That profligacy was punished in the 27th minute when, with the visitors' first genuine attack of the match, striker Craig McAllister was brought down by Magnus Okuonghae in the area.
McAllister stepped up and while Tony Roberts guessed correctly, the veteran keeper could only parry the ball back to the feet of the striker, who with the help of Ryan Harley bundled the ball into the net.
But Daggers hit back through Strevens. Ritchie threaded a ball through the defence and Scott Griffiths, celebrating his 200th appearance for the club, crossed to the back post where the striker was on hand to prod home.
However, the respite was short-lived as Exeter took just two minutes to regain the lead after the interval.
The energetic Steve Tully was released down the right and his centre found McAllister, who calmly swept the ball past the despairing Roberts.
The pattern of the game didn't change however, as the home side pushed hard for a leveller.
Visiting keeper Paul Jones flapped a cross onto his own bar, while Sam Saunders found the same spot with a late header.
But Exeter debutant Troy Archibald-Henville, signed from Tottenham Hotspur before the game, was imperious in defence as the Daggers' forwards toiled to no avail.