Ashley Grimes fired the hosts into an 18th-minute lead but the Shakers were better after the break and deserved their 51st-minute equaliser.
Phil Edwards was penalised for handling Adam Lockwood's shot and Shakers skipper Steven Schumacher beat Ben Smith at the second attempt before his side prevailed from 12 yards, with Trevor Carson decisively saving from Edwards. Earlier, Carson suffered a studs-up challenge from Craig Curran which saw the Dale forward receive an 80th-minute red card.
Coleman said: "I thought we were much the better team in the first half and, while we didn't really get out of the blocks in the second half Bury didn't really test out goalkeeper even though they had more possession.
"They had a big leg-up - we had three free-kicks given against us on the edge of our penalty area which should have been free-kicks to us. We had to defend those three set-pieces, and I've no idea what was going on.
"I can't criticise the referee, that would be churlish. I have to concentrate on my own players and we were comfortably the better side, and the best team has lost."