Luis Diaz’s clinical 13th-minute finish on his 100th appearance for the club since joining from Porto in 2022 set them on their way.
The goal came after a lightning break when a Brentford corner was cleared by Ibrahima Konate, Mohamed Salah set Diogo Jota away with Colombia forward Diaz finishing well.
Liverpool looked full of energy and Mark Flekken made two smart saves to deny Andy Robertson, while Brentford – who were once again without striker Ivan Toney as the transfer deadline approaches – had chances of their own.
Skipper Christian Norgaard spurned a glorious opportunity to equalise from a free header about 10 yards out which he put wide, while Alisson did well to keep out a header by Nathan Collins.
But Liverpool sealed the points when Salah clipped home with his left foot, the Egypt forward’s second goal of the season.
There is much to admire about this energetic Liverpool side under Slot and while there is certainly still room for improvement, they head to Manchester United next Sunday with back-to-back wins under the Dutchman.