The Premiership outfit won 34-12 to secure a bonus-point win and move within four points of pool leaders, Munster. Exeter drew with Munster, then lost to Castres away and Gloucester at home.
Exeter boss Rob Baxter added: "We've done the most important thing, we've given ourselves a reason to go to Munster and play what is close to a knockout game".
"It's positive that we focussed on our game and we worked hard to come through". That is the thing we have got to face first.
The Chiefs, though, still have hope after tries by full-back Jack Nowell, fly-half Joe Simmonds, lock Jonny Hill, wing Tom O'Flaherty, centre Henry Slade and replacement hooker Luke Cowan-Dickie - plus two Simmonds conversions - eliminated 14-man French champions Castres.
But their response was impressive, as far from going into their shells, they hit back with an equalising try just four minutes later, with Canada worldwide Paris rounding off a slick passing move to cross unopposed.
And it gave the Premiership belated cause for cheer, with five clubs - Bath, Wasps, Gloucester, Leicester and Newcastle - having been knocked out before the penultimate round of pool games had even been completed.
Castres' replacement prop Tudor Stroe was sent off midway through the second half for a swinging forearm into Chiefs centre Ollie Devoto's head.
Baxter paid tribute to England wing Nowell, who scored a sparkling solo try in the No 15 shirt on his first appearance since mid-November after being sidelined by a hamstring injury.
"That was Jack Nowell not 100% fit, which was quite nice to see, so hopefully he'll shake that off and a bit more game time will sharpen him up". When we are playing that relatively high-tempo game, it doesn't really matter what number is on his back.
The 25-year-old, playing at full-back, was a threat from the off, weaving his way through the visiting defence for the first try after barely two minutes.