The Gallagher Premiership leaders remain on course for the treble after a 48-14 victory over Exeter at Sandy Park.
England find themselves in a strange position heading into the final two rounds of the Guinness Six Nations.
Exeter picked up only their second Gallagher Premiership win of the season with a come-from-behind triumph against old adversaries Saracens.
Exeter Chiefs have confirmed the signing of Australia back-row Tom Hooper, as reported by RugbyPass.
Chiefs need a landslide bonus-point win in their final pool fixture against Ulster to have any chance of reaching the last-16, but an early exit still beckons because of their poor points difference.
Exeter Chiefs player ratings: Exeter endured a torrid evening at Sandy Park, as Toulouse delivered a 10-try masterclass to seal a thumping 64-21 win.
Round six of the Gallagher Premiership promises to deliver another thrilling weekend of action, with teams looking to solidify their positions ahead of a break before the Test rugby returns in November.
The Devon-based club, who have been part of the English top flight since 2010, have expressed alarm that steps could be taken to prevent fans from different clubs mingling freely at matches, including at Sandy Park.
They will assemble next Monday at Pennyhill Park before flying to Girona for a warm-weather training camp ahead of the November 2 opener in London versus New Zealand, a fixture followed by further home games against Australia, South Africa and Japan.