The victory earned Harlequins a 10th successive Aviva Premiership win.
Brown crossed 10 minutes after the break in between centre Matt Hopper's seventh-minute touchdown and flanker Luke Wallace's score as the leaders roared into a 22-6 lead with 57 minutes played at Adams Park.
But Hopper's departure for the sin-bin saw Hugo Southwell claim a first try for the home side and when England winger Tom Varndell followed suit nine minutes from time, the game was back in the balance.
However, stand-off Nick Robinson twice hit a post as he failed to add to his six points, and opposite number Nick Evans' single penalty and two conversions helped to see the visitors home for their 14th consecutive win in all competitions.
It took Quins just seven minutes to get their noses in front, Hopper running on to prop Joe Marler's inside pass to touch down beneath the posts, with Evans adding the conversion.
Robinson reduced the deficit within two minutes with a well-struck penalty and scrum-half Joe Simpson might have put Wasps ahead with when he broke through the cover and was grounded just feet short of the line by Hopper.
The Wasps fly-half should have dragged the home side back to within a point, but saw his penalty attempt come back off the post with the visitors finding themselves camped deep inside their own half.
A dominant home pack was causing all kinds of problems for Quins in the set-pieces, but they had to wait until the final minute of the first half to win the penalty from which Robinson pulled them back to within a point.
However, the Quins tight five restored pride as they turned the tables on their hosts at a 44th-minute scrum to allow Evans to stretch their lead to 10-6 four minutes after the restart.
They eased further clear in stunning fashion with their second try of the game as quick hands from number eight Chris Robshaw and Hopper gave the opportunity to Brown to chip over the top before winning the chase to touch down.
Quins cemented their lead seven minutes later when Wallace bludgeoned his way over from Danny Care's deft kick and Evans converted.
Hopper's afternoon took a turn for the worse when he was yellow-carded, and Wasps took advantage as Southwell touched down in the corner with 63 minutes gone.
Varndell made light of Hopper's return as he burst past him to run in his side's second try eight minutes later, although Robson hit the post for a second time with his conversion attempt to leave his side six points adrift.
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