Pat Lam's side knew that victory at Ashton Gate would secure a top-four finish for the first time since 2022.
They rose to the challenge with a dynamic display inspired by tries in the first five minutes from George Kloska and Joe Batley.
Kalaveti Ravouvou and Viliame Mata also touched down for Bristol in the first half, while AJ MacGinty added three conversions.
Two tries from Cadan Murley and one each for Alex Dombrandt and George Hammond, plus three Jamie Benson conversions, kept the visitors in contention.
But Gabriel Ibitoye bagged two second half tries before Santiago Grondona and Benhard Janse van Rensburg crossed over, with MacGinty making three more conversions.
In the play-off semi-finals, Bristol will face west country rivals Bath at the Recreation Ground on June 6.
Bath finished top of the Premiership, but fourth-placed Bristol have already beaten them twice in the league this season.
Leicester secured a home play-off semi-final with a 42-20 victory over bottom of the table Newcastle at Welford Road.
The second-placed Tigers, chasing a 12th league title, ran in six tries to book a date with third-placed Sale next Saturday.
Leicester trio Ben Youngs, Dan Cole and Mike Brown, who have over 300 England caps between them, now have one more match on home turf before retiring at the end of the season.
Sale made it to the play-offs after surviving a tense 30-26 win against second-bottom Exeter at Sandy Park.
Sale led 20-5 at half-time but Exeter got within four points with just over two minutes left before the visitors finally sealed the victory.
With top spot already wrapped up, runaway leaders Bath fielded a weakened team at Saracens and went down 36-26.
For only the third time since 2009, six-time English champions Saracens will be missing from the knockout phase of the competition.
Victories for Sale and Bristol meant a top-four finish was out of Saracens' reach as they finished a disappointing campaign in sixth place.
Having defeated Lyon in the European Challenge Cup final last weekend, Bath can now focus on finishing a memorable season with a double triumph.
Jack Cotgreave scored a hat-trick of tries as Gloucester clinched a bonus-point 41-26 victory against Northampton at Kingsholm.
Christian Wade added a brace with Seb Atkinson and Tomos Williams was also on the try-scoring sheet for Gloucester, with Santiago Carreras making three conversions.
However, the five points achieved were not enough for fifth-placed Gloucester to snatch a play-off spot.