Dons striker Michael Smith had put the home side in front midway through the first half.
In a tentative opening quarter-of-an-hour, Ross Worner saved comfortably from Daggers top-scorer Murphy, before Harry Pell swung in a teasing cross for the home side.
However, the Dons found the breakthrough in the 28th minute when Smith looped a header over Chris Lewington after a pin-point cross from Peter Sweeney.
The visitors nearly responded moments later as Zavon Hines' shot was saved by Worner after the former West Ham winger's mazy dribble.
Shortly after the interval, Dons winger Kaid Mohamed lashed over following a goalmouth scramble.
But after 58 minutes Murphy drilled a low shot from the edge of the area to draw the Daggers level.
The hosts had the best chances to clinch victory, but Smith drew a fingertip save from Lewington, while Pell flashed narrowly wide late on.