Play-off hopefuls Dagenham controlled the early stages but Bury hit back with two goals in five minutes from recalled defender Efe Sodje.
The Daggers then levelled after a quickfire double of their own, Jon Nurse scoring on the hour mark before Ben Strevens claimed a 64th-minute equaliser to secure a deserved point.
Sodje was making his first start in seven games following an injury to Ben Futcher while Dagenham were unchanged.
And it was the visitors that started strongest with Glen Southam testing Mark Tyler from 20 yards, before Tyler blocked a close-range strike by Paul Benson after Nurse squared from the right wing.
Bury were struggling to stamp their authority on the game and Benson looked to be clean through in the 23rd minute, but Sodje recovered to prevent the Daggers' striker getting a shot away.
Yet the home side snatched the lead following a left-wing cross by Dave Buchanan, with Paul Scott being snuffed out by Matt Ritchie only for Sodje to lash the loose ball under Tony Roberts from 12 yards.
And Sodje soon doubled Bury's lead after Aiden Palmer was booked for fouling Elliott Bennett on the right wing.
Brian Barry-Murphy swept in an inswinging free-kick and Sodje was left unmarked at the far post to head past Roberts from eight yards.
Solomon Taiwo tested Tyler with a curling effort just after the restart, while Benson sent a tame effort straight at the on-loan Bury keeper.
The hosts replied with Roberts saving Bennett's low drive and Andy Morrell hit a post from the rebound.
That proved to be a pivotal moment as the Daggers swiftly turned the game on its head, with first Nurse turning Buchanan before firing past Tyler from 12 yards.
Benson then got behind the Shakers' defence to square for Strevens and although his first effort was saved, he bundled home the rebound.
Both teams pushed for the winner, with Bury's Stephen Dawson going closest with a 20-yard shot wide.