Care is chasing a place in England's autumn internationals line-up
England scrum-half Danny Care scored a try and created two more as Harlequins maintained their bright start to the Guinness Premiership season.
Jordan Turner-Hall, Tom Guest, Gonzalo Tiesi and Mike Brown also touched down as Quins topped the fledgling table after their second successive victory.
They secured a bonus point despite fly-half Waisea Luveniyali scoring from only three of nine kicking attempts.
David Lemi scored Bristol's try, with Adrian Jarvis kicking two penalties.
Care had the visitors on the back foot after 12 minutes at The Stoop.
He took a tap penalty in his own half, launching a kick deep into Bristol territory and racing up to catch full-back Luke Arscott before he could find support.
Quins piled in to turn the ball over, moving it out for Turner-Hall to go over unchallenged.
Twelve minutes later, Arscott was in trouble again, dithering instead of clearing a long kick from opposite number Brown and allowing Brown to burst over.
Luveniyali's poor kicking display prevented Quins from going in at half-time with a commanding lead, with Jarvis, a close-season signing from the London club, reducing the deficit to 12-6 with two late first-half penalties.
The second resulted in the hosts going down to 14 men when left-wing Ugo Monye was yellow-carded for illegally preventing Bristol from re-cycling the ball.
Luveniyali's dire display continued when he failed with a 51st-minute penalty, but Bristol lacked both the invention and the penetration to trouble the home side.
Their hopes were effectively ended when Welsh international lock Robert Sidoli was sin-binned for preventing Care from releasing the ball.
Quins got their third try just before the hour mark. Luveniyali kicked for touch and, following the resulting lineout, Care scampered over.
The bonus point was secured six minutes later when replacement back row forward Guest scored another.
Care was the architect, scooping up a loose ball near the touchline to send Guest on his run to the line.
Bristol's defence cracked once more in the 75th minute when former England scrum-half Andy Gomarsall, on for Care, made a break to supply a try-scoring pass for Tiesi.
Bristol finally managed a try in the dying seconds when Lemi raced down the left wing, Jarvis adding the extras.
Harlequins rugby director Dean Richards: "We didn't quite play in the right areas of the field, and some of the decision-making at times wasn't good. But it's something we'll look at and we'll learn from it.
"That's the beauty about the youngsters. They are just so willing to learn. Some of the guys will be a little bit disappointed with their performance today, just from the intensity point of view, but we'll take the five points. We're happy with that.
"Danny's a handful and sometimes he over-eggs the situation, but there's no doubt England hold him in high regard as well as we do. I think he will there or thereabouts in terms of England.
"But I also thought Andy Gomarsall showed his quality when he came on today, and his experience steadied the side."
Bristol head coach Richard Hill: "It's hugely disappointing.
"A huge amount of work went into preparing this side, and we didn't see much product. We caused our own downfall in many respects.
"The first half was pretty dismal, and it's a shame people had to pay money to watch it. We made a whole host of errors all the way though. It isn't easy to explain when you watch that."
Harlequins: Brown; Strettle, Tiesi, Turner-Hall, Monye, Luveniyali, Care, Jones, Fuga, Ross, Kohn, Robson, Robshaw, Skinner, Easter. Replacements: Taione for Turner-Hall (72), J Evans for Care (66), Brooks for Jones (72), Botha for Fuga (60), Gomarsall for Kohn (72), Guest for Easter (60).
Not Used: Williams.
Sin Bin: Monye (39).
Bristol: L Arscott; T Arscott, Brew, Maggs, Lemi; Jarvis, Thomas; Irish, Linklater, Hobson, Sambucetti, Sidoli, Budgett, El Abd, Blowers. Replacements: Barden for Brew (69), Barnes for Maggs (60), Beveridge for Thomas (66), Bracken for Irish (62), Blaney for Linklater (66), To'oala for Sidoli (60), Ward-Smith for Blowers (52).
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