Coach Franck Azema praised "blossoming" Matsushima after the Japan full back took his season tally to seven in nine games as Clermont, a three-time runner-up, dominated the second-tier European Challenge Cup holder with some scintillating expansive play.
"We feel whether it's on or off the field in daily life he's blossoming. He feels good, he smiles, so we're happy. That's why we signed him," Azema said.
"We're happy that we're seeing more and more of him on the field. He's a big competitor in every sector, under the high ball and he's very efficient in his play."
Fiji's Semi Radradra missed Bristol's first match in Europe's top-tier club competition since 2008 with a leg injury suffered in last weekend's Autumn Nations Cup victory over Georgia.
The visitors were without Radradra's Test team-mate Peceli Yato as he was refused entry into Britain because of an administrative error.
South Africa-born Matsushima, who starred at last year's Rugby World Cup in Japan claiming five tries, opened the scoring after just three minutes.
He added his second 23 minutes after touchdowns by Apisai Naqalevu and Damian Penaud.
The French side led 29-14 at the break, already with the bonus point in the bag, as Max Malins and Bryan Byrne crossed for Bristol.
Penaud added a second and Fritz Lee slid over before Matsushima sealed his hat-trick with 12 minutes to go.
Bristol pushed for another bonus point for finishing within seven points as Harry Randall, Wales's Ioan Lloyd and Tonga centre Siale Piutau all went over, but ultimately failed to close the gap.
Clermont was one of four French teams to win away on the first weekend of the European Champions Cup.
In the late game, La Rochelle ground out a 13-8 win in Edinburgh.
Winger Raymond Rhule touched down after 24 minutes and full back Jeremy Sinzelle added a second try after 42 to put the visitors 10 points ahead. Dogged Edinburgh fought back with a try by Blair Kinghorn.
Four-time European champion Leinster, with Ireland five-eighth Johnny Sexton sitting the game out, proved too strong for Montpellier, running out convincing a 35-14 winner in France.
Flanker Dan Leavy crossed for Leinster's fourth try on 69 minutes to ensure a bonus-point victory. Full-back Jimmy O'Brien snatched a fifth at the whistle to confirm the Irish province's dominance.
Sale Sharks lost 26-14 at Toulon in its first game without head coach Steve Diamond, who resigned during the week.
There was more misery for English clubs as Bath lost 23-19 at home to the Scarlets.
Wasps, however, overcame Dragons in Newport 24-8.
On Monday (AEDT), champion Exeter Chiefs welcomes Glasgow Warriors with 16 players remaining from October's final win over Racing 92, which in turn hosts Connacht.