The draw extended manager Eddie Howe's unbeaten run since returning to Dean Court.
Tubbs' 76th-minute header cancelled out Greg Tansey's well-taken opener as the points were shared between two of the promotion contenders.
Bournemouth were looking for a sixth successive home win and announced the return of Bristol City striker Brett Pitman on loan before kick-off, but the Cherries never got going in the first half and were frustrated by some stubborn Stevenage defending.
Lewis Grabban's header from a Charlie Daniel cross, which almost caught visiting goalkeeper Steve Arnold by surprise, was the closest the hosts came to scoring before the break.
Stevenage took the lead in the 27th minute when Tansey beat David James with a powerful low shot from the edge of the penalty area.
Bournemouth improved in the second period, with Grabban passing up two good opportunities to equalise. The leveller arrived with just 14 minutes remaining as Tubbs headed in Grabban's cross from close range.