Kristian Dacey and Gareth Anscombe both crossed for the Blues while Steven Shingler's kicking sent them 15-0 up.
But Welshman Jordan Williams and Jack Tovey both scored tries to claw back 14 points for Bristol at the break.
Matthew Morgan scored either side of Williams' second try as Sam Warburton sent the hosts into the last eight.
The Blues, along with fellow Welsh region Ospreys, will not know their opponents in the quarter-finals until Stade Francais have faced off against Harlequins on Sunday.
The result of that game will decide who finishes second in pool five.
Cardiff Blues coach Danny Wilson told BBC Radio Wales: "What you saw in the second half is when we can build and sustain pressure and make teams kick poorly to us that lights up Matthew Morgan.
"We can play some fantastic rugby and you saw that with some of the tries we scored today."
Bristol's interim head coach Mark Tainton told BBC Radio Bristol: "For 65 minutes we played really, really well. We showed ambition against a very, very strong Cardiff Blues team.
"We just lost our way in the last 15 minutes by some inaccurate plays and inaccurate kicks. Making 11 changes, a lot of these guys haven't played masses of rugby this season.
"Ultimately, we're disappointed with what the score was in the end. The next five weeks are going to say a lot about the rest of our season."
Cardiff Blues: Matthew Morgan; Alex Cuthbert, Rey Lee-Lo, Steven Shingler, Blaine Scully; Gareth Anscombe, Lloyd Williams; Rhys Gill, Kristian Dacey, Anton Peikrishvili, Jarrad Hoeata, Macauley Cook, Josh Navidi, Sam Warburton (capt), Nick Williams
Replacements: Kirby Myhill, Corey Domachowski, Scott Andrews, James Down, Seb Davies, Tomos Williams, Garyn Smith, Rhun Williams.
Bristol: Luke Arscott; Jordan Williams, Jack Tovey, Tusi Pisi, David Lemi; Adrian Jarvis, Andy Uren; Jack O'Connell, Rob Hawkins, Gaston Cortes, Joe Joyce, Giorgi Nemsadze, Nick Koster, Jack Lam (c), James Phillips.
Replacements: Max Crumpton, Kyle Traynor, Jamal Ford-Robinson, Ian Evans, Nick Fenton-Wells, Martin Roberts, Thretton Palamo, Jack Wallace.
Referee: Mike Adamson (Sco)
Assistant referees: Sam Grove-White, Bob Nevins (Fra)
TMO: Patrick Thomas (Fra)
Cardiff Blues sealed their place in the European Rugby Challenge Cup quarter-finals as they survived a Bristol comeback at Cardiff Arms Park.