Aided by a red card for Bath's Quinn Roux just 13 minutes in, Josh Iosefa-Scott, Paul Brown-Bampoe, Len Ikitau and Campbell Ridl scored Exeter's tries in a bonus-point win to climb into fourth.
Dan Frost and Sam Underhill replied for the visitors, with Finn Russell slotting one of his two conversions.
But just a fourth defeat of the season cost them the chance to leapfrog Northampton at the top of the table.
Exeter open up a five-point lead on fifth-placed Bristol in the battle to likely join Northampton, Bath and Leicester in the play-offs.
"That was pretty special because there was a lot hanging on that game," said Exeter director of rugby Rob Baxter.
"They have controlled us pretty well over the last couple of seasons but today was a bit different, and our second-half performance into that wind was something special because that scenario was ready-made for Bath."
Harlequins thrashed Newcastle 76-17 in a clash between the bottom two.