Rovers dominated the first competitive fixture at their new £3.5million stadium, but slack marking from a 36th minute corner allowed Daggers man of the match Anwar Uddin to score the only goal with a bullet header.
As expected, Forest Green boss Gary Owers fielded a starting line-up which contained no fewer than 10 players making their competitive debuts.
An early season injury crisis saw Daggers manager John Still deprived of five first team players, forcing him to field midfielder Jake Leberl as a makeshift centre-half.
It was the hosts who started brightest, as a surging run from Les Afful in the opening minute saw him crudely brought down by Michael Olayle 20 yards out. The resulting free-kick, taken by Kevin Nicholson, flew into the side-netting with goalkeeper Tony Roberts beaten.
Dagenham responded four minutes later through Paul Benson, who beat full-back Alex Lawless for pace on the left flank and pulled the ball across for Glen Southam to poke a shot wide from close range.
Afful was looking sharp for Rovers, and his ninth minute cross from the right was flicked on and headed tamely at Roberts by Charlie Griffin.
Griffin's striking partner Allan Russell had an effort on 20 minutes, turning sharply in the left channel and sending an angled shot into Roberts' arms.
Ex-Dagger Alex Meechan produced an even better effort from 25 yards three minutes later, forcing an excellent save by the veteran stopper.
Rovers were dominating open play but were looking vulnerable from set-plays, so there was a sense of inevitability when Uddin rose highest to meet Sam Sloma's corner from the right and headed home from the penalty spot.
The home side's response was excellent, and both Russell and Griffin forced smart blocks from Roberts before half-time.
Dagenham's defence had to withstand even more pressure after the break, Russell firing over from the left edge of the area on 48 minutes before wasting a much clearer chance with a weak shot straight at Roberts eight minutes later.
The impressive Afful blazed over from 20 yards in the 64th minute, then an excellent pull-back by Russell somehow evaded the onrushing Griffin and substitute Mark Beesley - the pair colliding with each other from no more than six yards out.
Griffin is expected to score hatfuls of goals following his return to Nailsworth on loan from Wycombe Wanderers, but he will not be looking back on this match with any affection.
A long throw by Nicholson was flicked on by Polish replacement Arek Zarcynski into Griffin's path at the back post, where he scooped the ball over the bar from six yards.
And the popular striker ought to have rescued a point in the fourth minute of stoppage time, racing clear of a tiring Scott Griffiths and dragging his effort wide.
The visitors barely ventured into opposition territory during the second half, but they are due plenty of credit for hanging on to all three points and maintaining their impressive record at the Gloucestershire club.