Brown was struggling with his kicking on an unpredictable surface and he almost gifted Jimmy Ryan a similar goal, although he got back to recover and then also produced a wonder-save to deny Grant.
However, it all changed after the break. In the 53rd minute David Worrall fed Mike Jones who fired an unstoppable shot into the roof of the net to get Bury back on level terms. And two minutes later Shakers captain Stephen Dawson sent over a dangerous cross from the right with Stanley unable to clear and Andy Morrell applied the final touch.
Much to Alan Knill's frustration, though, it was all-square after 68 minutes when Symes' scuffed shot found Sean McConville at the far post and, with Bury appealing for offside, he fired the ball beyond Brown.
But Bury again came back in the 75th minute when Efe Sodje's header was superbly kept out by Reds goalkeeper Dean Bouzanis and Morrell got the last touch to make it a double. And Bury made it safe in injury time when Darran Kempson was adjudged to have fouled Ryan Lowe and Lowe tucked home the spot-kick.