John Dobson's side crossed for seven tries, with Jonathan Roche and Leolin Zas each scoring twice, to secure a bonus-point victory despite spending extended periods with a numerical disadvantage.
Hooker JJ Kotze received a 20-minute red card in the 33rd minute for a dangerous tackle, while Zas was sin-binned shortly after the restart for a high tackle. The Stormers led 21-7 at half-time but had to adjust their approach as Bristol attempted to capitalise on their advantage.
Dobson admitted his team had lacked energy in the opening half and had perhaps expected to win too comfortably after establishing dominance at scrum time.
"We were a little bit flat in the first half, energy-wise, and thought things were just going to come easily," he said.
The Stormers were forced to focus on game management rather than using their scrum and maul to wear down the opposition, but Dobson was encouraged by the leadership shown during that difficult period.
"We had to spend 25, 26 minutes in game management. Took a lot of fun out of the game. So we credit Paul (de Villiers) and the leadership, and then we've got it going."
De Villiers took over the captaincy after Ruhan Nel was ruled out shortly before kick-off, with Wandisile Simelane brought into the starting line-up at outside centre.
The 23-year-old flanker, who previously captained the Junior Springboks, said the experience of having Siya Kolisi and Deon Fourie alongside him helped ease the responsibility.
"I look to my left and there's a Springbok captain, and to my right there's Brannas, so it was an easy one to start with," he said.
While the result provided plenty of encouragement, Nel's injury is a potential setback for the Cape side. Dobson confirmed that the 35-year-old was suffering from a calf problem and hoped he would be available again within a week.
However, the Stormers will not rush their captain back, with Dobson warning that the injury could develop into a much longer absence if mishandled.
"Hopefully it is just a week," he said. "But we will obviously be conservative because that could be three or four months as well."
Dobson also highlighted Sacha Feinberg-Mngomezulu's accurate goal-kicking, with the flyhalf converting seven of his eight attempts.
"That's a magic kicking performance," he said. "It's good for Sacha's confidence. It makes a big difference on the scoreboard."
The Stormers now turn their attention to the United Rugby Championship, with Scarlets visiting Cape Town next week before Ulster arrive. The two home fixtures offer Dobson's team an opportunity to build on consecutive victories over the Sharks and Bristol.