The point was enough to keep Burton in ninth place in the table after they came from behind to lead at half time.
Nigel Clough's men made the worst possible start when goalkeeper Saul Deeney gifted the opening goal to the Daggers in the third minute.
The goalkeeper was guilty of hesitation as Ashley Vickers hooked the ball forward and Chris Moore capitalised with a deft ten-yard lob into the empty net.
Shane Blackett went close to increasing Dagenham's lead with a header, but the visitors were rocked by a spell of fierce Burton pressure and two clinical finishes.
Daryl Clare put the home team level with his first goal at the Pirelli Stadium since his move from Crawley last month.
The striker superbly controlled Gary Rowett's ball into the box before steering a ten-yard shot past Tony Roberts into the far corner.
Burton went in front in the 23rd minute, with Rowett again instrumental in opening up the visitors' defence.
They misread the flight of his free-kick and Andy Corbett ceased the chance to set up Shaun Harrad for a 12-yard shot into the bottom corner.
But Burton lost their way in the second half as Dagenham found another gear to pin them back for long periods.
Deeney had to be on his toes to save a 15-yard shot from Moore and a Vickers header.
The visitors were duly rewarded however in the 72nd minute when substitute Sam Saunders and Moore helped the ball on from a throw-in for Cliff Akurang to score with a close-range header.
And it was the visitors who went nearest to collecting all three points when substitute Paul Benson saw his header flash inches too high.