Wales flyer Alex Cuthbert crossed for the only try for the home side with international teammate Leigh Halfpenny converting and adding two penalties.
Cuthbert’s score gave the Blues what could be a vital losing bonus point but qualification is now out of their hands.
Lock Don Armand was Exeter’s scorer with Gareth Steenson booting 14 points to secure victory.
Halfpenny and Gareth Davies both missed late penalties for the Blues and Exeter saw Dave Lewis yellow carded but they did enough.
Armand’s early score shocked the Arms Park and with Steenson kicking two penalties and converting the try, Exeter were suddenly out in front. It was all the visitors as Rob Baxter’s pack set about getting on top. The Blues meanwhile were struggling to get a foothold in the game but all that changed when Cuthbert stood up.
Halfpenny kicked a penalty before the prolific winger finished off a move involving Halfpenny and flanker Ellis Jenkins, who was making his first Heineken Cup start.
With a three point deficit at the start of the start of the second half, Halfpenny soon drew the Blues level only for Steenson to hit straight back.
Halfpenny and Davies missed chances for the Blues and when Steenson landed two late kicks himself, it left the Blues wondering what might have been.
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