After a slow start at The Rec the Londoners took a stranglehold on the contest as they controlled the key areas at the scrum, line-out and breakdown, with Robshaw and Maurie Fa'asavalu excelling in their back row.
Robshaw's close-range effort had helped Quins to a 13-7 lead with Brown's solo efforts six minutes from time killing off hopes of a late Bath fightback. Fly-half Nick Evans contributed 16 points with the boot.
Bath's try came early on through lock Dave Attwood with Sam Vesty adding the conversion and two penalties.
Bath started brightly and Vesty missed two straightforward penalty chances inside the opening four minutes, hitting a post with his first before pulling the second wide from in front of the sticks.
Evans showed him how it was done when he gave the visitors a 3-0 lead after six minutes, but the hosts enjoyed the better of the early exchanges.
It took a brilliant double tackle from the Quins' duo of George Lowe and Robshaw to deny wing Tom Biggs as he burst over the line, a good tackle by Bath three-quarter Olly Woodburn on opposite number Sam Smith having yielded turnover ball on the visitors' 22.
But Lowe was at fault seconds later as his rash burst out of the defensive line allowed Attwood to bundle his way over for a try converted by Vesty.
The lead lasted only five minutes as Harlequins hit back with a try from Robshaw, the skipper showing good strength to muscle his way over having spurned the chance to go for goal from a penalty in the Bath 22.
Evans added the extras and quickly slotted his second penalty to give Conor O'Shea's side a 13-7 lead at the end of the first quarter.
The leaders were gifted a great chance to extend their advantage late in the half after a loose Evans' kick that went dead was adjudged to have been touched in flight by Vesty, handing Quins a five-metre scrum.
But strong first-up tackling saw Jordan Turner-Hall spill the ball and Bath survived.
Basic errors under pressure were hampering the hosts, and their failure to control the ball as they attempted to maul their way out of their 22 resulted in a 47th-minute penalty which Evans calmly slotted from out on the right.
Vesty responded with a penalty of his own to trim Bath's deficit back to six points, but a scuffle between the two sets of forwards saw the home side penalised and allowed the reliable boot of Evans to strike again.
The former All Black should have further extended Harlequins' advantage just before the hour mark when a purposeful surge from Jordan Turner-Hall presented him with a simple three-point shot, but he pulled his effort wide.
Bath boss Sir Ian McGeechan brought youngster Tom Heathcote on at fly-half and moved Vesty into the centre in an effort to change matters, while England scrum-half Danny Care made his return from three months out with a toe injury for Quins.
With time against them the home side set up a promising attacking position after a superb angle of running from captain Stuart Hooper, and Care was fortunate not to be yellow-carded after deliberately killing the ball in the shadow of his own posts.
Vesty needed the help of an upright to knock over the simple resulting penalty but Quins killed the game off as Brown, also the match winner against Exeter last time out, launched himself through a gap at the base of a ruck to sprint clear and round Nick Abendanon for a converted score, putting the seal on a win which sees them match Saracens' feat of two seasons ago.
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