James Short scored two tries in the last nine minutes as Exeter overcame an 11-point half-time deficit to beat Bath 17-11.
Two George Ford penalties and a try by Semesa Rokoduguni seemingly had Bath set for victory and they appeared well in control until a 62nd-minute Gareth Steenson penalty kick-started Exeter's challenge.
And Short crossed the line twice in the closing stages to move the Chiefs above Bath into third place in the Aviva Premiership.
Bath prop Nathan Catt, named in a 33-man England squad for a two-day camp in Brighton, lasted just 18 minutes before being led off for a concussion check and did not return.
Ford put Bath ahead with a short-range penalty in the 19th minute, scant reward for a strong start in which 19-year-old number eight Zach Mercer caught the eye.
A 45-metre penalty attempt by Steenson drifted wide, ending a run of 16 successful goal kicks by the Exeter captain.
Then an explosive break from his own 22 by Rokoduguni had the sell-out 14,509 crowd on their feet. Having caught a long punt upfield, he chipped the defence, leapt for the ball and brushed off three tackles before being hauled down just short of the line by Jack Nowell and Steenson.
Ford added three more points from a penalty awarded at the five-metre scrum.
Bath, with Ben Tapuai a creative force at inside centre, began to find their rhythm and Rokoduguni struck on 31 minutes with a well-deserved try that highlighted his finishing power at short range. Ford could not find the target from the touchline, though.
Exeter resumed with a greater sense of purpose but the Bath defence coped easily enough and returned to the attack, serving up a drop goal chance for Ford which he could not convert.
England wing Anthony Watson came off the bench for his first game in three months following a broken jaw. He was quickly into action, leaping for high balls and making a heavy tackle on Exeter full-back Phil Dollman.
Nowell was the most effective player in the Exeter back line, both in attack and defence, and his catch and break led to a scrum which earned a penalty on the hour, which Steenson slotted over from 40 metres.
The sheer attrition had now accounted for three of Bath's props as replacement loosehead Nick Auterac was led off to be replaced by a lock, Elliott Stooke.
Out of the blue, the Chiefs conjured a try in the left corner for Short after Luke Cowan-Dickie cleverly juggled a pass to the replacement wing.
Short struck again in the 77th minute after Bath ran out of defenders on the left, with Steenson taking his points tally for the game to seven by adding the conversion.