The game has failed to excite until Bruce struck in the 68th minute.
Receiving a throw from left-back Scott Griffiths, Tony Boot laid a perfect ball back to Bruce to fire past a helpless Matt Glennon in the Carlisle goal.
Desperate to find an equaliser, Carlisle poured forward and had a Danny Livesey shot cleared of the line by Dagenham's top performer Danny Foster in the 70th minute.
With the game opening up, Chris Moore had a chance to rap up the proceedings for the Daggers but he had a good shot saved.
At the other end, David Beharall headed over in the final minutes after good cross from substitute Karl Hawley.
In a largely forgettable first half, Carlisle will feel aggrieved not to have put at least one of their chances away.
Craig Farrell had two long-range efforts go off target - the closest coming just before half time.
Previous to that, a good one-touch passing move from the visitors saw midfielder pulled Paul Arnison have a one-on-one opportunity only for Tony Roberts to stand up and to save well.
The closest effort came after Boot fired wide when he really should have hit the target in the fourth minute.
With the back four on top form, Dagenham held on for a crucial victory and John Still's team continue to hunt for a play-off position.