Exeter recovered from an early red card and a 26-0 deficit to claw themselves level with just seven minutes remaining, but a last-gasp try from number eight Arthur Green saved Bath.
The drama began in the opening seconds when Henry Arundell was upended by opposite number Campbell Ridl and referee Luke Pearce showed the left wing a red card, reducing the visitors to 14 players for 20 minutes at the Recreation Ground.
Bath, fired up after their shock home defeat to Northampton last week, scored four tries in rapid succession through Beno Obano, Alfie Barbeary, Santi Carreras and Ollie Lawrence, opening up a 26-0 lead.
Once restored to 15 players, Exeter began to contest the breakdown more aggressively.
Hooker Joseph Dweba's try was quickly followed by a touchdown in the left corner by Olly Woodburn. Henry Slade converted both scores and Exeter were suddenly back in the game at 26-14.
Bath were now on the back foot, conceding a series of penalties that eventually forced them into desperate defence.
Led by their skipper, Dafydd Jenkins, the visitors surged to the home line and number eight Greg Fisilau plunged over for a deserved try in the 66th minute, but Slade's conversion was well wide.
Immanuel Feyi-Waboso squeezed in for a dramatic try in the corner, which Slade converted to bring the sides level.
Somehow, Bath dug deep to build a prolonged attack at the other end.
Carreras was hauled down under the posts, but replacement Green was in support to score before Finn Russell added the conversion - his fourth of the game.
The bonus-point win lifts defending champions Bath to 36 points - two clear of Exeter.
Northampton were hosting Harlequins in the later kick-off on Saturday.