It seemed the visitors were still suffering a Heineken Cup hangover when they conceded two tries in the opening 15 minutes, Ben Williams and Kyle Eastmond crossing for the rampant hosts.
However the Chiefs - who had leaked 36 unanswered points after the break against Clermont seven days earlier - threatened an almighty comeback in the second half.
Substitute Ignacio Mieres touched down with 12 minutes to go and with the already sin-binned Dan Hipkiss then sent off in bizarre fashion whilst on the sidelines, it appeared Exeter would complete the turnaround.
Bath, though, refused to suffer a first competitive loss to the Chiefs in 34 years, with Stephen Donald's late penalty sealing a thrilling win.
Exeter will look back on the inability of Gareth Steenson to kick two first-half penalties, while the No.10 also failed to convert prop Brett Sturgess' try from next to the touchline.
Yet in truth they were fortunate to only be 15-5 behind at the break, such was Bath's dominance. Williams burst through two weak tackles to break the deadlock after just three minutes, then Eastmond soon followed him in diving over in the right corner to finish off a flowing team move.
They also scored the third try of the contest in the 58th minute, the excellent Francois Louw showing a neat turn of pace for a flanker to squeeze in out wide, albeit completely against the run of play.
Bath then had to hold on, their cause not helped when Hipkiss was shown a second yellow for interfering with the ball after it had been kicked off the pitch.
Donald finally gave them some breathing space with three late points, while Exeter missed a late, late penalty that would've gained them a losing bonus.
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