The returning Jack Nowell scored the opening try before Joe Simmonds, Jonny Hill and Tom O'Flaherty all crossed to secure a bonus point for the Chiefs.

Taylor Paris and Thomas Combezou scored either side of half-time for Castres.

But there were further tries from Henry Slade and Luke Cowan-Dickie as Castres played the last 20 minutes with 14 men after Tudor Stroe was sent off.

Exeter must beat leaders Munster at Thomond Park next week to have a chance of winning their pool, or qualifying as a runner-up and joining Saracens as the only other English club in the last eight.

Nowell inspires Exeter

Nowell returned from a two-month injury lay-off in spectacular fashion with the opening try. Playing at full-back, he collected the ball 35 metres out, danced past a couple of tacklers, and sprinted clear of the remaining Castres defenders.

Castres hit back straight away as Paris went over in the corner after their only spell of pressure in the first half as Benjamin Urdapilleta's excellent pass bypassed the home defence.

But Simmonds scored his first try of the season, after Ollie Devoto evaded a challenge and put in a perfect pass for the young fly-half to go in from five metres out - and Hill went over under the posts from close range after relentless pressure.

Bonus point gives Chiefs a chance

Winger O'Flaherty, who has taken his chance after injuries to Nowell, Alex Cuthbert and Olly Woodburn, scored a well-deserved try soon after the restart, and although Castres hit back, the result never looked like being in doubt.

Slade and Cowan-Dickie both went in from close range as Castres' outside hopes of getting back into the game evaporated when Stroe was sent off after replays showed he recklessly swung an arm at an Exeter player in the ruck.

And the win could mean Exeter's poor start in the competition is forgotten if they can go to Munster and get another bonus-point victory and top a pool they looked like being left behind in.

Nowell, 25, has played 74 matches for Exeter scoring 125 points

'We had multiple ways of hurting them' - what the coach said

Exeter director of rugby Rob Baxter told BBC Sport:

"We have been waiting for the injection of real quality rugby, real energy, and I thought we saw that. Our attack looked a lot better, multi-faceted.

"We were finding spaces around the edges of rucks, we were finding space back on the inside, the momentum we could gain with some of our carrying was good, our clearout work was light years better than away at Castres when we struggled at breakdown."

On having a chance to go through at Munster:

"We've done the most important thing, which is get to the last stage of the pool in a semi-knockout game. We've given ourselves something to go over there and play for.

"We can genuinely talk about going and testing ourselves in one of Europe's iconic venues against one of Europe's iconic teams in as close to a knockout game as you can get, and it's going to be great to see which players want to stand up and fight."

Exeter: Nowell; Cordero, Slade, Devoto, O'Flaherty; Simmonds, White; Hepburn, Yeandle (captain), Francis, Dennis, Hill, Skinner, Armand, Kvesic.

Replacements: Cowen-Dickie, Moon, Williams, Lees, Lonsdale, Maunder, Steenson, Whitten.

Castres: Palis ; Paris, Combezou, Ebersohn (captain), Smith ; Urdapilleta, Radosavljevic; Tichit, Jenneker, Kotze, Lassalle, Jacquet, Gerondeau, Delaporte, Vaipulu.

Replacements: Firmin, Stroe, M. Clerc, Mafi, Jelonch, Tulou, Vialelle, Kockott.