Lowly Bristol looked set to prevent Exeter from going back to the top of the table when they held a 14-point advantage in the second half of a thriller at Ashton Gate.
Big-money signing Piutau took a quick tap and dropped his shoulder to barge his way over for a five-pointer on his Premiership bow and 14 points from the boot of Callum Sheedy put Bristol 19-12 up at the break.
Moray Low and Tom Lawday went over in the first half for Exeter, but John Afoa whipped a superb no-look pass for Luke Morahan to dive over in the corner and Sheedy converted to put Bristol 26-12 ahead within six minutes of the restart.
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Rob Baxter’s men were never going to give up their 100 per cent record lightly, though, and Santiago Cordero took a pass from Gareth Steenson before stepping inside to dart past three players and touch down.
Jack Yeandle added another try 10 minutes from time as the Chiefs ramped up the pressure and Bristol had to do without Piutau – reportedly the highest paid player in the league – after he was left with a bloody nose.
Sheedy had seen a penalty strike the post before Steenson booted Bristol into a five-point lead with five minutes to go, but Exeter won it when a penalty try was awarded after a charging maul was brought down in time added on.
FULL TIME: Bristol Bears 29 Exeter Chiefs 31
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