City dominated the early stages and took the lead after 18 minutes. Adam Barton produced a 25-yard screamer after good pressure from City's front three - a feature of the first half - and a nice exchange of passes with Carl Baker.
Cody McDonald created a number of good opportunities and on the occasion he was stopped illegally, Baker stepped up to smash home the resultant free-kick from 19 yards in the 28th minute.
Bury got back into the game 10 minutes into the second half when debutant David Healy expertly-dispatched a penalty, after he was clumsily brought down by Richard Wood.
The Shakers grew into the game after that and were back on level terms after 69 minutes.
Joe Skarz took advantage of the space he was given on the left-hand side to produce a fine cross for Healy's replacement Lenell John-Lewis to head home.