Drewe Broughton saw an early header headed off the line and in the home side's next raid Michael Proctor and Chris Llewellyn refused openings in front of goal.
Sam Aiston also shot wide and the Daggers escaped when their keeper made a rare mistake dropping a dangerous cross which was scrambled away.
The visitors began to push upfield as they gained confidence. David Rainford shot wide from 30 yards and Ben Strevens nearly gave his side a surprise lead when he pounced on a loose ball and shot inches off target.
Simon Spender and Matt Done tested Dagenham's defence with dangerous crosses as it looked likely that Wrexham would score.
Broughton headed just off-target and Aiston missed from close range. Wrexham keeper Gavin Ward dived low to make a good save from Peter Gain and saw a header from Strevens fly narrowly past his left post.
Roberts saved brilliantly from Aiston, then dived to save a Broughton header which went out to Proctor, whose shot hit the post and rebounded clear.
Broughton missed another chance from 15 yards and substitute Neil Roberts, pushing up from midfield, headed wide when it seemed he must score from point-blank range.
Substitute Stuart Nicholson also headed wide from a Simon Spender cross.
In a tense few minutes near the end both sides showed signs of nerves with vital points at stake and Wrexham counted themselves fortunate to finish level, with the possibility of doing well enough to escape relegation still there if they begin to hit the target.