Strikes from Stewart Drummond and Ryan Williams ensured victory for Jim Bentley's side who trailed through a 61st-minute Luke Wilkinson header.
Wilkinson gave the home side the lead from close range from a Matt Saunders free-kick. The central defender jumped higher than Nick Fenton to head the ball past Barry Roche from six yards out.
The Shrimps then hit back in dramatic fashion. Drummond levelled the scores with nine minutes remaining when he ran on to Jack Redshaw's superb through ball before firing a dipping right-foot shot past Chris Lewington from 12 yards out.
The comeback was complete two minutes later when substitute Williams scored a superb individual goal. The new signing showed great feet in the box to create space for himself before clipping a left-foot shot past the goalkeeper.
Luke Howell almost earned the Daggers a point with the final kick of the game with a shot that grazed the right-hand post with Roche beaten.