Only stoppage time remained and the Swans held out comfortably for a point from a game they dominated for periods, particularly in the second half.
The first half was slow to ignite, Swansea having plenty of the ball and looking a threat without managing a serious goal attempt in the opening half hour.
Bristol were just as ineffective going forward and after 29 minutes their head coach Lee Johnson made a change, sending on striker Famara Diedhiou for centre-back Nathan Baker. Tommy Rowe dropped from left wing-back into the back-three.
Moments later Andreas Weimann fired over from distance and Diedhiou went close to breaking the deadlock in the 39th minute, sliding in to connect with a Callum O’Dowda cross and seeing his effort graze the far post.
Both sides had looked to attack in the opening 45 minutes, but there were too many misplaced passes where it mattered.
Swansea began the second half on the front foot, Bersant Celina forcing a save from Bentley with a well-struck free kick from the left.
Still the visitors looked to be positive and Celina volleyed over from the edge of the box. By the 58th minute Bristol had used all three replacements.
Still it was Swansea posing the greater problems as Jay Fulton lifted a volley over from inside the box. Home fans were getting restless and reacted with jeers when Andre Ayew headed just over from a Grimes corner.
Bristol were under incessant pressure. Ayew broke clear on the right and Bentley had to advance to make a brave save. Then Mike van der Hoorn headed over from a corner.
The hosts were struggling to defend set-pieces and another corner from the right saw Joe Roden nod wide.
At the other end Diedhiou headed straight at goalkeeper Woodman. Brownhill had a late free-kick deflected over, but a point was the post Bristol merited.