The South African side ran in seven tries and secured a bonus-point win in their Pool 3 opener, showcasing sharp attacking rugby and clinical finishing against a largely second-string Bristol outfit.
The Stormers struck early as Seabelo Senatla crossed in the seventh minute, followed quickly by another try by Leolin Zas. Jonathan Roche added a third before half-time as the home side built a 21-7 lead.
Bristol's only first-half response came from a Joe Owen try, converted by Sam Worsley.
The visitors briefly narrowed the gap after the break when Sol Moody scored on his Champions Cup debut, but the Stormers quickly reasserted control.
Roche crossed for a second try, and the Stormers kept piling on points as the Capetonians' pace and offloading game proved too much for the Bears' defence.
Fly-half Sacha Feinberg-Mngomezulu was influential with the boot, converting all seven tries.
The Stormers' backline, featuring Damian Willemse, Senatla and Zas, created consistent problems, while the forwards, including Springboks Siya Kolisi and Evan Roos, provided a strong platform.
Bristol, rotating their squad with one eye on the Premiership campaign, showed flashes of resistance but struggled to contain the Stormers' attacking patterns for long.
A 20-minute red card for Stormers hooker JJ Kotze after a high tackle on Bristol captain Kieran Marmion, and a yellow to Zas early in the second half, gave the visitors a temporary numerical advantage, but they couldn't capitalise and it was the Stormers who finished the match in emphatic fashion.