Cheered on by a vast away following, the Shakers took a 35th-minute lead through Mike Jones before Brian Barry-Murphy struck just three minutes later with his first goal of the season.
Keeper Wayne Brown then gave Bury a huge boost with his third penalty save of the season to deny John Mullin on the stroke of half-time.
Sub Jamie Clarke had better luck as he pulled one back for Stanley with a 79th–minute spot-kick, but the Shakers deservedly held on to go back up to sixth in League Two.
The home side made a positive start with on-loan striker Greg Blundell firing high and wide after being put clean through, before Jimmy Ryan fired over in the 10th minute.
Bury then began to get a foothold in the game and Andy Bishop had a goal disallowed for offside after linking up with strike partner Glynn Hurst. Stanley striker Paul Mullin then tested Brown from the edge of the box before Bishop chipped narrowly over after being played in by Hurst.
But Bury opened the scoring shortly after the half-hour mark when Barry-Murphy found Bishop in the right channel and he crossed for the unmarked Jones to head home with ease at the far post.
The Shakers soon doubled their lead when Hurst picked out Barry- Murphy’s burst from midfield and the Irishman took one touch before slipping the ball past home keeper Kenny Arthur.
Stanley had a great chance to cut the deficit when Efe Sodje was penalised for tripping Blundell but John Mullin’s spot-kick was too close to Brown.
Bury were content to stand firm in the second half and Rostyn Griffiths’ long-range effort was the best Stanley could muster until Clarke fired straight down the middle from the spot after pushing in the box.
Elliott Bennett and sub Dean Howell had earlier gone close to sealing it for Bury but two goals proved to be enough for Alan Knill’s men.
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