The home side freshened-up their starting line-up after least week's disappointing 30-8 defeat away to Leicester Tigers, as Australian lock Dean Mumm made his debut at the expense of skipper Tom Hayes.
Hooker Simon Alcott made his first Premiership start while flanker Ben White, fly-half Gareth Steenson and centre Phil Dollman were the other changes while No.8 Richard Baxter led the side.
The visitors had made two changes from the side that lost to Saracens last week, with Karl Dickson starting at scrum half while loosehead prop Mark Lambert made his first start of the season.
Steenson was quick to make an impression on the game with a good break and then the former Ireland Under-21 landed a 25 metre penalty but his opposite number, Nick Evans, saw his attempt to quickly level the scores drift wide of the uprights.
Quins maintained the pressure with Matt Hopper linking-up with his centre partner Jordan Turner-Hall but the England international was stopped short. Chiefs knocked-on but at the scrum the home side were penalised for Evans to get his side back on level terms.
Chiefs crossed for the first try of the game after a break from full back Luke Arscott which, after several phases of play, saw Dollman give a long pass to prop Brett Sturgess out on the touchline who ran round behind the posts.
Streenson added the extras and a penalty but Evans kept Quins in touch with his second penalty, from 48 metres as the Chiefs led 13-6 midway through the half. Evans then charged down Steenson's clearance kick to touchdown.
Evans failed with the touchline conversion but following Exeter pressure Steenson landed a second penalty, however England wing Ugo Monye went over for his fourth try of the season with Evans adding the conversion to give the visitors an 18-16 lead at the break.
Exeter made the perfect start to the second half with blindside flanker Ben White going over from close range, as referee Martin Fox continued play after a questionable tackle by Mike Brown on Hopper, and Steenson's conversion took the Chiefs back in front.
Midway through the half, wing Matt Jess saw his pass intercepted by replacement fly-half Rory Clegg who raced up to the Chiefs line, chased by Jess, before passing inside to scrum half Karl Dickson with Clegg converting.
Exeter, sensing a possible upset, responded with two tries in three minutes through inside centre Jason Shoemark to secure a try-scoring bonus point, with Steenson converting the second for a 10-point lead.
The home side were forced to defend their line as the game went in to the final 10 minutes with some strong tackles. But they then gave away a penalty and Clegg landed his second kick to give his side a possible losing bonus point.
But the Chiefs sealed the game with two minutes remaining as replacement Sireli Naqelevuki broke clear for Exeter's fifth try and Steenson converted. Munn was yellow carded in the final minute but his side held on for the win.
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