The Shakers sealed all three points thanks to goals from midfield pair Michael Jones and Brian Barry-Murphy on 35 and 38 minutes at the Fraser Eagle Stadium.
Stanley midfielder John Mullin then saw a penalty saved by Wayne Brown in the last minute of the opening period after new signing Gregg Blundell had been felled in the box by Efe Sodje.
Jones broke the deadlock when he nodded home from close range at the back post following persistence from Andy Bishop on the right side of the area.
Glynn Hurst then headed the ball into the path of Barry-Murphy shortly after and the Irishman coo'ly slotted past Kenny Arthur from just inside the 18-yard box.
Brown then preserved Bury's lead with his third spot-kick save of the season from Mullin's weak effort.
Sub striker Jamie Clarke pulled one back for Stanley from the spot with 11 minutes left - Paul Mullin the man fouled - but it was too little too late.
Accrington had marginally been the better of the two sides before the opener in the East Lancashire fog.
Loan signing Blundell fluffed a couple of opportunities, while there were constant doubts about whether the game would be completed due to the inclement weather conditions.
Paul Mullin was believed to have had a second-half header cleared off the line for Stanley, although it could not be confirmed due to the lack of visibility.
Bury held out fairly comfortably in the second half, although it was made harder than it should have been when Clarke fired home from the spot.