The win sees Harlequins to a comfortable mid-table position, while Bristol find themselves still hovering close to the relegation zone just third from bottom in the table.
It was the home side who opened the scoring, former Ireland international fly-half kicking a penalty in the sixth minute.
But Bristol struck back just four minutes later with a magnificent long-range try from David Rees, the wing latching onto a loose Quins pass and charging 70 metres upfield to score.
The Londoners grasped the initiative some eight minutes later through Burke’s second penalty and then the fly-half showed the other dimension to his play by seizing on a loose ball and making a break before giving a scoring pass to team-mate Ben Gollings.
But it did not take Quins long to reassert their authority. Fly-half Burke converted another penalty before setting up replacement Ben Gollings to score. Burke converted to give Harlequins a 13-5 lead at half-time.
Bristol came out of the dressing room firing on all cylinders, former All Black Daryl Gibson showing great power in the 50th minute to score his side’s second try.
Felipe Contepomi was on hand to land the conversion and while Burke was able to claw back a further three points through a third penalty goal, Bristol again struck with Rees scoring his second try of the afternoon.
But an error by wing Brendon Daniel was to cost Bristol dearly in the closing stages of the match.
The former Quin was unable to touch down the ball over his own line after Will Greenwood had nudged a purely speculative kick downfield, thus allowing the England centre to dive on the ball for a soft try. With Burke converting, Harlequins were able to hold on for victory.
NEC Harlequins:Tries: Ben Gollings, Will GreenwoodCons: Paul Burke (2)Pens: Paul Burke (4)
Bristol Shoguns:Tries: Daryl Gibson, David Rees (2)Con: Felipe Contepomi