Robins defender Aden Flint gave the hosts the perfect start inside 10 minutes, bundling Jake Bidwell's left-wing cross into his own net.
But former Brentford defender Karleigh Osborne levelled for the visitors two minutes later, firing across Bees goalkeeper David Button into the far corner before immediately limping off with a hamstring injury.
Tyrone Barnett should have given the visitors the lead minutes later, but volleyed wide from close range with the goal at his mercy.
On-loan Blackburn midfielder Alan Judge made City pay midway through the first half when his blistering drive from 12 yards after a slick counter-attack was too hot for goalkeeper Elliot Parish to handle.
Marcello Trotta made it 3-0 just before the break, sweeping home Judge's low cross to help record Brentford's 10th home win on the trot, their best sequence since 1936.
Brentford shut up shop after the break and despite dominating the only noteworthy chance fell to Barnett, whose rising drive flew over from the edge of the box.