Fabinho opened the scoring just before the interval for the Reds, converting with a close-range header at the back post from a corner.
Alex Oxlade-Chamberlain doubled the lead on 69 minutes when he diverted an Andy Robertson cross home with his head.
Takumi Minamino came off the bench to claim a third after Roberto Firmino had regained possession from a Brentford goal kick.
The result moves Liverpool up to second in the standings, leapfrogging Chelsea with a game in hand on the Londoners.
