Substitute Marlon Jackson then netted late on to leave the visitors waiting for their first league win.
Each side hit the woodwork early on, Sinnott's cross clipped the crossbar before Accrington's Will Hatfield struck a post from 12 yards.
Bury went ahead after 28 minutes when Craig Jones played a free-kick short to Sinnott, who chipped the ball in for Forrester to steer home at the far post.
Stanley were up against it five minutes later when Webber was sent off for a late challenge on Shaun Beeley. And the hosts doubled their lead after 38 minutes when Jones' cross was cleared as far as Sinnott, who beat one man before firing home.
Despite their numerical disadvantage, James Beattie's men were on top for most of the second half and Rob Atkinson went close when he headed narrowly over. But they lost Nicky Hunt to injury before Jackson scored on the break from Jones' 87th-minute cross.