The Shakers extended their winless run to four games as Joe Jacobson headed home a Sean McConville corner in the 77th minute.
It was a deserved winner for Stanley as they had been thwarted by Bury keeper Cameron Belford, who made fine saves from Terry Gornell, Jimmy Ryan (twice) and McConville, throughout.
"We didn't do ourselves justice there (at Gigg Lane) and I am just pleased we did here," said Coleman.
"I am delighted for the players because they were down after losing a two-goal lead to Chesterfield on Monday and drawing 2-2. I know I spat my dummy out but the players were hurting too and they have made up for that.
"They have some good players but we limited them and we were value for the win. Now we have got to try and do something we haven't done this season and get back-to-back wins. We are a good side but sometimes we do ridiculous things and throw points away - we held firm until the final whistle."