Brown, who has played for Quins since 2005, was one of five try scorers at the Stoop as the hosts claimed their fourth Premiership win in a row for the first time since 2014.
Joe Marchant scored his 11th try of the campaign, while young star Nathan Earle grabbed his seventh try in the league to cement the win late on.
Both sides entered the game in contrasting form, with Quins having picked up 20 points in their last six league matches while Bristol had won just two of their last nine.
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The Bears made the better start and dominated possession in the opening exchanges, resulting in Charles Piutau powering over the line after a kick from Callum Sheedy set-up Steven Luatua to penetrate the home defence.
In the topsy-turvy opening 20 minutes, Bristol responded well with them applying pressure in the scrum and Alapati Leiua was able to dive over in the nick of time to hand back the advantage to the visitors.
The home side rebounded yet again and Marcus Smith saw a gap in the Bristol defence and, thanks to a dummy, he was able to weave through their defence and go over.
In an action-packed first half, a slick move of passes between Alex Dombrandt, Elia Elia and Chisholm on the left-hand flank allowed Brown to go over for his record-equalling try for Harlequins in the 32nd minute.
With less than five minutes to go before the half-time whistle, Marchant gave his side the bonus point after an unselfish pass from Ben Glynn saw him ghost through to give the home side a 14-point advantage at the break.
The Bears kept on attacking and Piutau’s weaving run allowed replacement Andy Uren to pass to Sheedy, who went through under the posts and give Bristol hope of a comeback.
However, Bristol gave away a penalty with little over 10 minutes to go and James Lang sent the kick over from 40 metres for the home side.
Earle then took advantage of a stunning pass from Dave Lewis in the 77th minute to dot down before Bristol earned a consolation try through Leiua’s second try of the match.
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