The visitors would have been unfortunate to be trailing at the break, let alone 2-0 down, after missing a host of first-half chances.
Yet that was the half-time scoreline after Brian Barry-Murphy and Andy Morrell struck for the Shakers in the last four minutes of the first half.
Scott Davies pulled one back with seven minutes left but Bury hung on to go third in League Two.
Alan Knill's men made a lively start with in-form winger Mike Jones cutting in from the left and forcing Shots goalkeeper Mikhael Jaimez- Ruiz to parry a long-range strike.
Morrell then headed over from Elliott Bennett's right-wing cross, but Shots responded with Bury's on-loan keeper Mark Tyler having to make two fine saves to deny Scott Davies and Rob Elvins from the resulting corner.
Elvins headed over moments later after Hudson had got to the right byline, while Davies fired narrowly wide as the visitors continued to pose the Shakers' defence problems.
Morrell might have done better with a header from a right-wing ball by Jones, who had swapped wings with Bennett, and Bury were soon back on the defensive.
Hudson latched onto a long ball, forcing Tyler to charge out of goal, but Bennett's last-ditch tackle prevented a certain goal, while Elvins fired wide after Shots again got behind the Bury defence.
Yet it was the Shakers that struck first, Barry-Murphy bundling the ball past Jaimez-Ruiz after playing a one-two with Andy Bishop.
Rhys Day then failed to deal with a low right-wing cross by Bishop and Morrell pounced to stab home his first goal in four months.
Morrell fired wide shortly after the restart while Bishop headed over before Tyler did well to hold onto a low strike by Anthony Straker.
Grant turned a Hudson ctross wide late on before Davies fired home after Tyler had been penalised for picking up a backpass by Stephen Dawson.
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