The play was far from textbook but Quins had enough to do the double over the Chiefs in front of a sell-out crowd of 10,602 at Sandy Park.
The visitors had gone in to the break 8-6 in front but replacement fly-half Gareth Steenson looked to have sealed the game six minutes from time with a 40 metre penalty.
But England Saxons fly-half Rory Clegg, who had a nightmare of an afternoon with his kicking as he missed the target with four penalty attempts and a conversion, landed his second penalty to clinch the game three minutes from time.
Both sides went in to the game after post-Christmas defeats with the Chiefs suffering their first loss in five games as they were beaten 29-22 at London Irish while Harlequins lost their first Premiership game of the season as they went down 19-11 to defending champions Saracens at Twickenham Stadium.
There was a new look to the Chiefs front row of Brett Sturgess, Neil Clark and former Quins' player John Andress, Aly Muldowney came in to the second row while the sole change among the backs saw Jason Shoemark recalled at inside centre.
Quins will be without former All Black Nick Evans due to a minor injury so Clegg started at fly-half while England's Nick Easter missed the game due to an ankle injury so Tom Guest started at number eight.
The visitors looked dangerous from the start and made the Chiefs pay for their errors with scrum half Danny Care taking a quick tap penalty before reaching for the line for opening try which was awarded after consultation with TMO Ed Morrison.
Clegg missed the touchline conversion and a penalty attempt from a similar position five minutes later but his opposite number Ignacio Mieres showed that he could find the target when a Chiefs catch and drive was pulled down.
But Quins continued to apply the pressure with a good break with the ball going through four pairs of hands but 10 metres from the line the pass from full-back Mike Brown went to ground midway through the half.
Clegg had another opportunity to increase the Quins advantage, with 12 minutes of the half remaining, but his kick from out on the right hit the near upright only for Mieres to put the Chiefs in front minutes later from just inside the visitors' half.
Exeter continued to take control of the game and began to pin Quins back in to their own half as half-time approached.
But the visitors' defence held firm and Clegg finally found his mark with a penalty on the stroke of half-time to give Quins an 8-6 interval lead.
The Chiefs started the second half strongly and had the upper hand in the scrums and the loose play as they forced the visitors to defend but when they did break into the Exeter half they were awarded a 25 metre penalty which Clegg saw sail past the far upright.
He had another opportunity a couple of minutes later, this time from the right, and again he failed to trouble the touch judges.
Exeter continued to pin the visitors back into their own half with some excellent breaks.
But Exeter, when in a commanding position 20 metres form the line, choose the wrong option as scrum half Hayden Thomas went on the narrow side and chipped ahead for wing Gonzalo Camacho only for the ball to be collected by Quins wing Ugo Monye.
Steenson, who had been on the field for just three minutes having replaced Mieres, saw his penalty attempt from halfway fall short.
Then from 10 metres closer the touch judges were not sure if Steenson's next effort was good.
Referee JP Doyle called on Morrison to adjudicate and he decided that the ball had gone over the bar to put the Chiefs in front with six minutes to play.
But Clegg, who had blown 14 points, restored the Quins' lead with three minutes remaining with a kick which proved decisive.
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