A stoppage-time penalty by substitute Phil Jevons was enough to take 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.
Stanley should had of taken the lead in the 58th minute when substitute Chris Turner won a penalty from virtually his first touch.
Brown appeared to get something on the ball as he dived at Turner's feet, although Peter Cavanagh fired the spot-kick against the top of the crossbar.
That gave Bury a boost but Jevons missed a gilt-edged chance with 20 minutes left, firing wide from a knock-down by fellow sub Andy Morrell. Andrew Procter was sent off for handball in stoppage time and although Jevons slotted home the spot-kick, time ran out for the Shakers.
League Two play-off semi-final fixtures: First legs: Shrewsbury v Bury, Rochdale v Gillingham (both Friday 7 May).
Second legs: Bury v Shrewsbury (Sunday 10 May, 12pm), Gillingham v Rochdale (Sunday, 10 May 6.30pm).