The Chiefs claimed their first appearance in a European quarter-final in style at Sandy Park.
Exeter looked destined for a heavy loss as the French club surged into a 14-0 lead within 10 minutes of kick-off but turned the game around thanks to the boot of Ignacio Mieres and a Hayden Thomas try.
Brett Sturgess and James Phillips then grabbed second-half touchdowns to seal the win and qualification to the knockout stages.
But Exeter will have to wait until Wasps' clash with Bordeaux-Begles on Sunday to learn if they have done enough for a home quarter-final.
The Catalan giants came flying out of the blocks at Sandy Park with quickfire tries from Charles Geli and Jeffrey Michel, both converted by James Hook.
Mieres hit back with the boot before Thomas burrowed over following a driving lineout.
Sturgess raced clear to score on the hour before Phillips wrapped up the win, cutting through after Gareth Steenson's penalty rebounded off the post.
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.