Harlequins levelled the score as Cunningham-South bulldozed his way through between the posts for his first ever Harlequins try following some quick passing between Danny Care, Andre Esterhuzen and Dombrandt.
Genge then sold a brilliant dummy to run behind to dot down and open up a seven-point lead for Bristol.
The prop then got through for a third Bristol try but it was disallowed for a foul against hooker Thacker on the ground in the build-up.
A television match official review showed Dombrandt grounded ball in the right-hand corner after a sustained period of Quins pressure finally got over the line, following some resilient defending from their opponents, with Marcus Smith converting from distance.
But Bristol were again in the lead moments later as Ibitoye took advantage of a mistake to charge through between the posts unchallenged.
The visitors were dealt a blow when Genge and Sheedy both left the field looking to be in real discomfort towards the end of the first half.
However, the Bears began the second half the same way they started the first, with Thacker scoring inside three minutes to secure the bonus point and open up a 12-point lead, before Williams hit the post with the conversion attempt.
Northmore turned over the ball on the break and kicked and chased before juggling over the line to reduce the deficit to five points.
Cunningham-South ran through for his second try as Quins took the lead for the first time and also gave them a bonus point, in what would be the high point of a disappointing afternoon for the hosts.
Yet Williams opted to kick three points from a penalty to regain the lead for Bristol and set up a tense last 20 minutes.
Their fifth try from replacement Lahiff was met with an almost stunned silence at The Stoop as the home crowd saw their semi-final hopes fade and Williams kicked another penalty to open up an 11-point lead.
Winger Heward left the Quins defence in his wake to charge through for a sixth try in an incredible second-half Bristol performance.
Replacement Oghre added a seventh as Harlequins were unable to take advantage of their stacked bench as the game got away from them.