Full-back Brown crashed through the Chiefs defence to secure Quins' ninth consecutive win
League leaders Quins were pushed all the way by Exeter before securing their ninth straight win in all competitions.
Despite going 9-0 down at the break to the boot of Nick Evans, Exeter were far from outclassed and were twice thwarted close to the Quins line.
Chiefs took a 13-9 lead after a move which started in their own 22 was rounded off by Bryan Rennie.
But the try sparked Quins into life and heavy pressure ended with Mike Brown going over after a ruck to win it.
Quins struggled to cope with the strong Exeter pack in the first half and had to call upon all their defensive nous to keep the Chiefs at bay.
The hosts were reduced to 14 men when lock George Robson marked his 100th appearance for the club with a yellow card for an off-the-ball incident.
Chiefs failed to make the most of their numerical advantage but eventually took the lead through Rennie who went over after Matt Jess had been held up two metres out.
A regalvanised Quins at last went on the offensive and a surging attack ended with Karl Dickson's off-load sending full-back Brown crashing through the defence for a fine try.
Exeter remained undaunted and forced a series of scrums close to the Quins line but the hosts held out after what had been their stiffest challenge of the season.
POST-MATCH REACTION
Harlequins rugby director Conor O'Shea:
"We had to dig in out there, and to come back as we did and score 10 quick points after falling behind speaks volumes for the attitude of the players.
"It was not the prettiest of games because there was a lack of tempo, but then we realise that teams will come here to stop us playing."
Exeter Chiefs head coach Rob Baxter told BBC Radio Devon:
"I think most of the stats will say that we should have won.
"I've got some frustrations with some areas out there, but ultimately you've got to look beyond that.
"Making it about the officialling doesn't improve you as a side. What does is looking a the things you can control."
Replacements: Tatupu for Shoemark (44), Barrett for Thomas (16), Moon for Sturgess (75), Clark for Whitehead (59), Tui for Andress (46), Bentley for Hanks (75), White for Johnson (63). Not Used: Steenson.
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