The high-flying hosts controlled most of the opening period but were stunned by two strikes in five minutes from Dawson and Jones, who both claimed their first goal for the club.
Elliott Bennett then struck late on to ensure the Shakers bounced back to winning ways after surrendering their unbeaten league record last week.
Bury had the game's first chance with Andy Bishop curling wide from Bennett's through-ball but the Daggers soon took charge.
Sam Saunders and Paul Benson were off-target from outside the box while Ben Strevens had a glancing header beaten out by Wayne Brown from a Saunders corner.
Bury snatched the lead in the 32nd minute, though, as Dawson dispossessed Peter Gain before Bishop played in Dawson from the loose ball and the Irish midfielder showed great composure to chip the onrushing keeper.
Gain tested Brown with a superb 25-yard strike but Jones replied with one of his own, which caught Tony Roberts off-guard and flew past the home keeper.
Strevens missed a gilt-edged opportunity to pull one back from Benson's cross, although the hosts posed Bury few problems after the restart.
Jones saw a 59th-minute strike deflected against a post before David Buchanan got back well to deny home substitute Mark Nwokeji a shot at goal.
Saunders curled a free-kick narrowly wide heading into the closing stages while Solomon Taiwo fired over.
But Roberts denied Bishop before Bennett fired home eight minutes from time from sub Chris O'Grady's lay-off, making Saunders' stoppage- time strike merely a consolation.
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