Champions Exeter moved 11 points clear at the top of the Aviva Premiership after Joe Simmonds' late penalty thwarted Bath 20-18.
The young fly-half booted half of his team's points, while his older brother -- England number eight Sam Simmonds -- scored a try, along with Chiefs lock Jonny Hill.
Bath looked to have done enough for a crucial victory in their Premiership play-off push when Wales international fly-half Rhys Priestland, making his first appearance after injury for two months, converted substitute Paul Grant's try 13 minutes from time.
Bath had trailed by 11 points just 15 minutes earlier, yet they displayed deep reserves of character and commitment to fight back, with wing Matt Banahan claiming a well-worked try and Freddie Burns kicking two penalties.
But ultimately they were denied, with a losing bonus leaving them four points behind fourth-placed Newcastle and games beginning to run out.
England head coach Eddie Jones looked on as Exeter quickly moved into their impressive close-quarter driving game, which created a platform for Simmonds to kick them ahead through a sixth-minute penalty.
Bath struggled to get going and could find no answer to Exeter's sustained intensity, with the Chiefs proving masters at keeping possession.
Simmonds sacrificed a kickable penalty to turn the screw by kicking to touch inside Bath's 22, and the home side could not escape as the Chiefs patiently put together phase after phase in pursuit of an opening try.
And it duly arrived after 23 minutes as a dominant Exeter pack laid siege to Bath's line, and Hill claimed a try that referee Wayne Barnes awarded following confirmation from the television match official Rowan Kitt.
Simmonds added the conversion to open up a 10-point lead, but Bath then enjoyed the best spell of their half, securing a period of territory and keeping Exeter pinned back, even if some of their attacking play was laboured.
Two Burns penalties in nine minutes cut the gap though, and Exeter trooped off at half-time with only a 10-6 lead, having seen Bath rewarded for some overdue front-foot rugby led by number eight Taulupe Faletau and flanker Matt Garvey.
But Bath were immediately rocked back during a punishing opening nine minutes of the second period as Exeter went through 24 phases of play inside the home 22 before Sam Simmonds touched down from close range and Joe Simmonds converted.
Bath had defended superbly to keep the Chiefs out, yet Exeter's heavy-duty approach had to take a toll, and they led by 11 points with 30 minutes of the contest remaining.
Bath had to respond quickly, and Burns obliged by sending a pinpoint kick rolling into Exeter's 22 that Banahan capitalised on by grasping a kind bounce and putting his team back into contention.
The score lifted Bath's game across the board, and Exeter, despite rugby director Rob Baxter making a number of changes off the bench, could not initially respond as Faletau's sharp midfield break saw him deliver a 67th-minute scoring pass to Grant.
Priestland's conversion edged Bath ahead, but Exeter then laid siege to the opposition 22 once more, and Simmonds' late penalty clinched it.