In the last attack of the game Molesley cut inside from the right flank. He then lined up a super left-footed shot that beat the dive of Tony Roberts. The ball nestled in the back of the net off a post right in front of a delighted travelling support.
In an otherwise uninspiring game the Daggers had the better of the first half and the visitors were grateful to Warren Cummings for a goal-line clearance that prevented Sam Saunders from opening the scoring for the home side.
A super save from Shwan Jalal also prevented Doug Loft from getting his first goal for the Daggers.
Danny Foster had the ball in the back of the net but that was ruled out for offside while a Saunders free-kick skimmed the post as the home side continued to be frustrated by a determined Bournemouth defence.
Matt Ritchie had a couple of long-range efforts that failed to bring any reward for the Dagenham winger during a second half that saw more ambition from the visitors.
Sammy Igoe stretched the Daggers defence and chipped over a cross that found Jeff Goulding at the far post. The sub very nearly scored only to see his header rebound off a post.
It was the Cherries who looked more likely to win the game as the final whistle approached. Goulding had another go near the end after beating Roberts to the ball, the forward however couldn't get his shot on target from the left touchline.
It didn't matter though for the Cherries as Molesley's late strike gave them three priceless points.