A see-sawing battle at Sandy Park was eventually won by visiting Bath thanks to the boot of George Ford and three tries.
The Chiefs' run of good form had come to an end at the hands of Northampton on their last outing, while Bath's had continued with a win over Newcastle.
The two teams traded penalties early in the first half and there was little between the two teams for the majority of the game.
The game's first try was a joy to behold, and it went to Bath. George Ford ghosted through the defensive line before finding Kyle Eastmond, who laid the ball on for Ollie Devoto to finish the move.
Ford added the conversion to make it 3-10, but ten points from Exeter in the final moments of the first half handed the lead back to the hosts.
After Gareth Steenson's penalty goal, the TMO awarded a try to Ben White, who was adjudged to have grounded the ball under a pile of bodies.
Steenson added the conversion to make it 13-10 to Exeter at the break, and they went six points up after the restart when Steenson hit the target with a third penalty goal.
But two tries in ten minutes from Bath midway through the second half turned the game on its head. Eastmond finished off a fantastic team move, which Ford converted, to put Bath back in front, before Nick Abendanon showed great individual skill to chip the ball and regather for a sensational try.
Ford was again on target with the extras to make it 13-24, but Exeter hit back almost immediately when Ian Whitten broke through a stretched Bath defence for Steenson to convert.
Bath led by a point heading into the last ten minutes, and they were able to preserve their narrow advantage in the period that followed, before sealing victory through Ford's penalty goal in the final minute.
Tries: White, Whitten
Cons: Steenson 2
Pens: Steenson 3
Tries: Devoto, Eastmond, Abendanon
Cons: Ford 3
Pens: Ford 2
Exeter: 15 Luke Arscott, 14 Matt Jess, 13 Ian Whitten, 12 Phil Dollman, 11 Tom James, 10 Gareth Steenson, 9 Dave Lewis, 8 Dave Ewers, 7 Ben White, 6 Tom Johnson, 5 Damian Welch, 4 Dean Mumm (capt), 3 Hoani Tui, 2 Jack Yeandle, 1 Ben Moon.
Replacements: 16 Luke Cowan-Dickie, 17 Carl Rimmer, 18 Alex Brown, 19 Don Armand, 20 Kai Horstmann, 21 Haydn Thomas, 22 Henry Slade, 23 Fetu'u Vainikolo.
Bath: 15 Anthony Watson, 14 Horacio Agulla, 13 Ollie Devoto, 12 Kyle Eastmond, 11 Matt Banahan, 10 George Ford, 9 Micky Young, 8 Leroy Houston, 7 Francois Louw, 6 Matt Garvey, 5 Dave Attwood, 4 Stuart Hooper (c), 3 David Wilson, 2 Ross Batty, 1 Paul James.
Replacements: 16 Eusebio Guinazu, 17 Nathan Catt, 18 Anthony Perenise, 19 Dominic Day, 20 Alafoti Fa'osiliva, 21 Peter Stringer, 22 Gavin Henson, 23 Nick Abendanon.
Referee: Andrew Small
Assistant referees: Roger Baileff, Gareth Copsey
Television match official: Keith Lewis