The home side became the first team to post a perfect record in the pool stage this season as they piled on the points to underline their qualification for the quarter-finals.
They scored nine tries in all, with Cuthbert crossing twice in the opening half before completing his treble in the last 10 minutes.
Bath are one of three English clubs through to the quarter-finals, joining Wasps and Gloucester, and will be the top-ranked team with 29 points. The five pool winners will be joined by three runners-up from the Heineken Cup in the last eight.
Gloucester also made it six from six as they ran in six tries in their 36-16 home win over Mont-de-Marsan.
England hopeful Billy Twelvetrees came off the bench to grab a second-half penalty and convert the final try by Jonny May, who was one of two Gloucester players to cross twice, with Henry Trinder following suit. Charlie Sharples contributed a try in the 22nd minute.
London Irish and Worcester ended their pool campaigns on a high with home victories, but neither side was able to finish higher than second.
Jonathan Joseph and Marland Yarde went over for the Exiles, while Steve Shingler kicked seven points.
Worcester ran in 11 tries in a 71-19 win over Spanish champions Bizkaia Gernika. Outside-half Danny Gray helped himself to 21 points with eight conversions and a try.
Wales scrum-half Mike Phillips was among the try-scorers in Bayonne's 54-0 win over Rugby Mogliano in Italy, while Grenoble hooker Laurent Bouchet grabbed two of his side's tries in a 47-0 win over Cavalieri Prato.
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