A double strike by Daggers' hit man Craig Mackail-Smith was enough to derail the Railwaymen at Glyn Hopkin Stadium.
Mackail-Smith headed in his first of the afternoon straight after the break from a Glen Southam cross.
His second came when, after pursuing Southam's clearance, he held off defender Andy Roscoe's challenge and coolly slotted the ball past Paul Crichton.
The Daggers should have capitalised on their majority of possession in the first half and were guilty of squandering chances through Chris Moore and Mackail-Smith. The two, however, did combine well after 20 minutes only to see Mackail-Smith's shot tipped over by Crichton.
As the interval approached, the Railwaymen had chances through an Alex Meechan header which went straight at Tony Roberts, while the best chance of the half fell to Karl Rose. Graham Hockless and Rose linked up on the right, Rose then skipped past the advancing Roberts but saw his effort cleared by the covering Lee Flynn.
Just before the hour mark, Southam, Mackail-Smith and Tony Boot all combined with Southam just failing to connect. Railwaymen were unlucky when Hockless and Meechan exchanged passes after a corner, but Barry Miller hit the bar from Meechan's cross.
Late into the game the visitors continued pressing. Gareth Stoker and Chris Lane both had shots saved by Roberts. Daggers were comfortable in the end, their first clean sheet of the season, but Leigh have real worries.