The win was Exeter's first victory over Saracens at Sandy Park and the visitors' first defeat of the season.
Sireli Naqelevuki scored the game's only try after 54 minutes, with the rest of the Chiefs' points coming from three first-half Ignacio Mieres penalties in a narrow 14-12 success.
All of Sarries' points came from the boot of fly-half Owen Farrell - two penalties in each half - but his efforts were not enough to help his team avoid a first loss of the campaign.
Saracens went into this match unbeaten in the Premiership this season, having drawn 9-9 against Leicester at Wembley last weekend and having won their last four games on the road.
The visitors included England international Chris Ashton in their starting line-up, the 25-year-old donning a head bandage after suffering minor cuts to the left side of his face during a brief alteration with a stranger during a night out in London last weekend.
Scrum-half Neil de Kock made his 150th appearance in a Sarries side led by flanker Kelly Brown. Hooker Jamie George, lock Alistair Hargreaves and outside Duncan Taylor all made their first starts of the campaign.
Exeter opened their season at Sandy Park with a convincing 43-6 triumph over Sale but that was followed by defeats at Northampton and newly-promoted London Welsh.
Injuries to Craig Mitchell and Tom Johnson meant prop Carl Rimmer and flanker James Phillips came into the pack. Scrum-half Haydn Thomas was handed his first start of the season, with Ian Whitten moving to the centre and Phil Dollman going to the wing.
Both teams took time to settle into the game and adjust to the slippery ball before the hosts went close to scoring after 10 minutes when Thomas charged down a kick from full-back Andy Goode but, after several drives, the Chiefs knocked on at the line.
But having given away several penalties, a couple when in attacking positions, at the breakdown it was Saracens who broke the deadlock as fly-half Owen Farrell landed a 25-metre penalty from just to the left of the posts.
Just before the midway point of the half the Chiefs were back on level terms thanks to a 40-metre penalty from Argentina fly-half Mieres, 10 metres in from the right touchline.
But the Chiefs failed to learn from earlier errors and were again penalised at the breakdown as Farrell slotted a 25-metre kick from in front of the posts.
Hargreaves was penalised soon after but Mieres' 30-metre kick hit the upright.
Just before the half-hour mark, with the Chiefs pressing the line, Sarries skipper Brown was shown the yellow card by referee Sean Davey and Mieres kicked the resulting penalty and his third a couple of minutes later to put Exeter 9-6 in front at the break.
Farrell, now matching Ashton with a head bandage after having treatment off the field in the first half, levelled the scores with his fourth penalty shortly after the restart before Ashton was sent to the sin-bin for a shoulder-charge on James Phillips, but Mieres failed with the kick.
Exeter took advantage of the extra man with inside centre Naqelevuki taking an offload from wing Matt Jess to score in the corner only for Farrell reduced the arrears with his fourth penalty just before Chiefs hooker Chris Whitehead was yellow-carded on the hour.
Both sides continued to mount attacks in the final minutes, with the visitors striving for maximum points. Sarries replacement fly-half Charlie Hodgson kicked for touch rather than attempt a long-range penalty, but it came to nothing.
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