Connolly's second-half double took his tally to four goals in three games and moved the former Premier League side off the bottom of the League One table at the Shakers' expense.
After a dire first half the match exploded into life four minutes into the second half.
Youngster Jed Wallace cut infield from the left before feeding the unmarked Connolly, who with his left foot slid the ball between the legs of goalkeeper Trevor Carson.
Connolly's second followed 15 minutes later as captain Johnny Ertl's shot forced a good save from Carson and the Irishman showed good composure to take the ball onto his left foot before coolly slotting home.
Bury striker Craig Fagan should have given the visitors the lead early on when he found himself clean through with only goalkeeper Simon Eastwood to beat, but he somehow blazed the ball into the stand from six yards out.