A Sears breakaway created another opportunity for the visitors, as his cross found Ryan Fraser who headed straight at the Brentford keeper.
Sears tested Button with a long-range strike for the visitors, before the Tractor Boys took the lead on 45 minutes through a Bru volley.
Ipswich doubled their lead after the break, when Sears drew Button towards the ball before finding Fraser for a tap-in.
Bru tested Button again soon after with a low shot from the edge of the area, before substitute Andre Gray attempted to chip Bialkowski to score for Brentford, but his effort went over the bar.
Bialkowski was forced into another save late on to deny Lasse Vibe from close range, before Brentford found a goal through Andre Gray on 92 minutes.
The Bees found an equaliser on 95 minutes, when Tarkowski tapped in from close range after a defensive mix-up by Ipswich.