That win for the home side means a win against Munster in Thomond Park could be enough to send them into the quarter-finals, while Castres will host Gloucester Rugby.
But the Gallagher Premiership leaders still face a Herculean task to progress, needing victory over twice European champions Munster in Limerick next Saturday.
"When we're playing a relatively high-tempo multi-phase game it doesn't really matter what number he's got on his back".
"We can genuinely talk this week 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 as we could get". That is the thing we have got to face first.
Tries by Jack Nowell, Joe Simmonds, Jonny Hill, Tom O'Flaherty, Henry Slade and replacement hooker Luke Cowan-Dickie - plus two Simmonds conversions - saw off 14-man French champions Castres. Castres replacement Tudor Stroe was also sent off in the latter stages for foul play.
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 global Paris rounding off a slick passing move to cross unopposed.
La peculiar "estrategia" del Milan para retener al 'Pipita — Gonzalo Higuaín
La cuestión es que el delantero no estaría a gusto en el conjunto rossonero y buscaría cambiar de aires en el próximo mercado. Vamos a ver qué es lo que sucede con Higuaín .
And although wing Taylor Paris and centre Thomas Combezou scored tries for Castres, with fly-half Benjamin Urdapilleta adding one conversion, Exeter were good value and deservedly maintained their European dream.
"We spoke about having a fast start and we did that", Nowell, returning to action with a try after suffering a hamstring injury on England duty during the November internationals, told BT Sport.
"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".
"Jack can play pretty much anywhere across the backline".
"There is certainly a possibility of him staying there (full-back) for a little while now, after playing so well".