Referee Greg Macdonald awarded the Chiefs the late seven-pointer, which sealed a crucial win that takes them back above Saracens at the summit.
The visitors’ other try scorers were Moray Low, Tom Lawday, Santiago Cordero and Jack Yeandle, with Gareth Steenson kicking two conversions.
Charles Piutau and Luke Morahan crossed for Bristol but it was the boot of Callum Sheedy that provided most of the Bears’ points, 19 to be precise.
Bristol, who move up to 10th spot, put in a spirited first-half showing to go into the interval 19-12 up, thanks to a try from Piutau and Sheedy’s boot.
Sheedy opened the scoring in the fourth minute with a penalty after a strong carry from Will Hurrell resulted in Exeter Chiefs offending at a ruck.
Four minutes later though the Chiefs were over through prop Low, who finished off superb work from Lawday to cross from close range for 7-3.
The Bears though, through Sheedy, added points on the board again on 15 minutes as this time Don Armand was penalised for hands in the ruck.
Bristol were now growing in confidence and when a high tackle from Mitch Lees on Sheedy resulted in a third three-pointer, they went 9-7 up before the fly-half made it 12-7 on 27 minutes. That fourth penalty goal from wide on the right side arrived after Exeter held on at a ruck.
Exeter though hit back in style on the half-hour mark as a kickable penalty was sent to the corner, Lawday the player finishing off that crossing.
However, it would be the Bears who had the final say of the half as Piutau showed strength and smarts to tap quickly and put his side ahead.
Bristol came out for the second-half with purpose too and quick hands from John Afoa set up Morahan for their second try that made it 26-12.
But that buffer didn’t last long as a brilliant response from Exeter that saw them rack up the phases led to Cordero stepping inside for 26-19.
Bristol then came agonisingly close to making it a 10-point lead but Sheedy’s long-range penalty hit the bar, which sparked an Exeter onslaught.
It took until the 69th minute for the Chiefs to get reward for their efforts when a penalty went to the corner and a slick move saw Yeandle reach out, with Steenson’s missed conversion leaving Exeter 26-24 down.
That became 29-24 on 76 minutes when Sheedy struck after a scrum offence, but then Exeter threw the kitchen sink at Bristol late on, with a driving maul being taken down and referee Macdonald marching under the posts.
Tries: Piutau, Morahan
Cons: Sheedy 2
Pens: Sheedy 5
Tries: Low, Lawday, Cordero, Yeandle, Penalty try
Cons: Steenson 2
Bristol Bears: 15 Charles Piutau, 14 Luke Morahan, 13 Piers O’Conor, 12 Will Hurrell, 11 Ryan Edwards, 10 Callum Sheedy (cc), 9 Andy Uren, 8 Nick Haining, 7 Dan Thomas, 6 Steve Luatua (cc), 5 Aly Muldowney, 4 Ed Holmes, 3 John Afoa, 2 Shaun Malton, 1 Yann Thomas
Replacements: 16 Tom Lindsay, 17 Jake Woolmore, 18 Lewis Thiede, 19 Joe Joyce, 20 George Smith, 21 Harry Randall, 22 Tiff Eden, 23 Sam Bedlow
Exeter Chiefs: 15 Phil Dollman, 14 Santiago Cordero, 13 Ian Whitten, 12 Ollie Devoto, 11 Alex Cuthbert, 10 Gareth Steenson, 9 Stuart Townsend, 8 Tom Lawday, 7 Matt Kvesic, 6 Don Armand, 5 Mitch Lees, 4 Ollie Atkins, 3 Greg Holmes, 2 Jack Yeandle (c), 1 Moray Low
Replacements: 16 Elvis Taione, 17 Billy Keast, 18 Marcus Street, 19 Wilhelm van der Sluys, 20 Dave Dennis, 21 Nic White, 22 Joe Simmonds, 23 Sam Hill
Referee: Greg Macdonald
Assistant referees: Jonathan Healy, Steve Lee
TMO: Stuart Terheege