A stoppage-time penalty by substitute Phil Jevons took the Shakers level with Wycombe, who surprisingly lost at home to Notts County.
But Alan Knill's men were unable to grab the second goal they needed to take them above the Chairboys and will continue their promotion bid against Shrewsbury in the play-offs.
Accrington refused to lie down for Bury and Craig Lindfield flashed an early shot across the face of goal.
The Shakers looked nervy and took time to settle, Elliott Bennett firing their first effort over from just inside the box after a run inside by Mike Jones, who then dragged a 25-yard effort well wide.
Bennett then chipped the ball into the six-yard box and Kieran Charnock did well to head clear.
Stanley replied with John Miles curling wide from just outside the box before Glynn Hurst spurned the best chance of the first half in the 23rd minute.
Bennett put Hurst through with an inch-perfect slide-rule pass but the veteran striker dragged his first-time strike just wide of the far post.
Nerves seemed to set in again as neither side went close until Peter Cavanagh volleyed over for the visitors from a Miles corner just after half-time.
Miles then forced a save from Wayne Brown after latching on to Sean McConville's ball inside and waltzing past two Bury defenders.
Chris Turner then replaced the injured Bobby Grant and won a penalty from virtually his first touch after Accrington broke from a Bury corner.
Brown appeared to get something on the ball as he dived at Turner's feet, although Cavanagh fired the spot-kick against the top of the crossbar, sparking wild celebrations amongst a bumper home crowd.
Alan Knill attempted to provide a further boost by bringing on both Jevons and Andy Morrell up front with 20 minutes left and Jevons fired wide from just five yards from Morrell's knock-down.
Jones then had a low cross cleared while Dave Buchanan chipped the ball narrowly over in a frantic finish.
Kenny Arthur saved late headers from Efe Sodje, Ben Futcher and Danny Racchi as Bury surged forward.
The Shakers were then awarded a penalty, with Andrew Procter sent off for handball, and Jevons slotted home the spot-kick.
That sparked a pitch invasion by the home crowd but 1-0 was not enough for Bury and Premiership referee Andy D'Urso controversially blew the final whistle after appearing to play barely two of the five minutes of stoppage time that had been indicated.