A 13-minute burst at the start of the second half at Sandy Park proved crucial.
The visitors were quick off the mark at the restart, scoring 15 points to regain the lead - which they held on to and increased without reply.
Exeter will question the services of the TMO as their England flanker Tom Johnson appeared to have tackled Clermont Auvergne's centre Wesley Fofana - who went on to score a try in the corner early in the second half which changed the face of the game.
The Chiefs, last season's losing Amlin Challenge Cup quarter-finalists, named an unchanged starting line-up from the side that lost 9-6 away to Heineken Cup champions Leinster in Dublin last weekend.
Clermont were forced to make one change after their Scotland international lock Nathan Hines was banned for six weeks, so French international Julien Pierre came in to the second row alongside Jamie Cudmore.
Hines received the ban from an independent disciplinary hearing for stamping on Scarlets' wing Morgan Stoddart after being cited for the incident, which happened during the first half of their 49-16 Heineken Cup victory at Stade Marcel Michelin last Saturday.
Exeter dominated the early passages of play and fly-half Gareth Steenson put the home side in front in the third minute with a 48-metre penalty and then the Irishman added a second from 25 metres on 10 minutes.
The visitors then came into the game but France international scrum-half Morgan Parra, who kicked 17 points last week against the Scarlets, was off-target with a 30-metre penalty and then wing Napolioni Nalaga had a try ruled out by TMO Kevin Beggs.
But after a poor Chiefs lineout the visitors seized the ball and the Fijian international squeezed in the corner with Parra adding the touchline conversion to put Clermont Auvergne in front.
Exeter then showed that they were prepared to pursue running rugby and, after 25 minutes, the visitors were again penalised for going off their feet and Steenson landed a 41m penalty to restore the the advantage for the Devon side.
The visitors were then penalised for wheeling a scrum 10m from their own line and Steenson landed his fourth kick.
Exeter again squandered another opportunity when in an attacking position and then when they were then forced to defend there was a yellow card for number eight Ritchie Baxter for obstruction. Parra reduced the arrears to two points at the break with a penalty.
Clermont Auvergne looked dangerous in the early stages of the second half against 14-man Exeter and full-back Luke Arscott and Ian Whitten were forced to bundle Nalaga in to touch just short of the line.
The visitors regained the lead after five minutes of the half through a Parra penalty followed by Wesley Fofana breaking from 30 metres to score, despite the video replay showed a foot in touch after a Tom Johnson half-tackle, which Parra converted.
A second try followed soon after as Wales international full-back Lee Byrne gave the scoring pass to flanker Julien Bonnaire for a 25-12 lead. But Exeter, like when they are behind in the Premiership, were not prepared to let their heads drop.
As the game went into the final 10 minutes Clermont Auvergne went in search of a try-scoring bonus point with New Zealand international wing Sitiveni Sivivatu, converted by Parra, followed immediately by a second from Fofana which David Skrela converted.
Just before the final whistle Exeter's replacement fly-half Ignacio Mieres was yellow carded for a deliberate knock-on. From the tap penalty Nalaga went over for his second try with Skrela adding the extras.
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