Teenager Afolabi Obafemi and Rhys Murphy struck either side of the interval looked to have the Daggers in cruise control as they led the struggling Shakers by two goals.
However, Anton Forrester scored from the penalty spot to reduce the deficit in the 89th minute, and then the woodwork came to the hosts' rescue as Bury's Gareth Roberts fired an effort against the post deep into stoppage-time.
It meant Bury's miserable away record goes on. They've only bagged one point on the road so far this season, and are now without a win in five league and cup matches.
"Some of the lads are shattered, they've given it everything but the ball just wouldn't fall for us at the end," admitted Blackwell.
"I felt we had the better of the first half, but when you don't take your chances you can get hit on the counter-attack. That's exactly what happened towards the end of the first half.
"It was a terrible time to concede a second too, not long into the second half.
"But we've shown some character, though it was far too late to make a difference.
"We've given ourselves far too much of a mountain to climb and that's something we've got to cut out of our game.
"We've got a hard week's work in front of us now."