Daggers made three changes for the match, with Jake Sheppard, Andre Boucaud and Corey Whitely coming into the starting line-up.
Daggers almost took the lead in the 14th minute when Luke Guttridge crossed for Oliver Hawkins from the left, but the striker’s half-volley from the penalty spot went just wide.
Eleven minutes later, Daggers took the lead in slightly fortuitous circumstances through Corey Whitely.
The forward crossed from the edge of the area, but his effort went all the way in and past a helpless Scott Davies in the Tranmere goal.
With five minutes left in the half, Elliot Justham was at full stretch for Daggers to keep out a 20-yard shot from James Wallace.
Daggers doubled their lead in the 47th minute when Oliver Hawkins lashed home from 18 yards after a good run.
Seven minutes later, Hawkins saw a header from a corner cleared off the line by Tranmere captain Steve Jennings.
Despite mounting pressure from Tranmere, Daggers were able to hold on and secure a vital away win.