The win ends a three-match winless patch in the league for Liverpool that has seen them fall off the pace in the title race.
Liverpool were without Mohamed Salah and Sadio Mane who are playing at the Africa Cup of Nations and their lack of potency in attack was evident as they struggled to open up a determined Brentford defence.
The home side finally broke through a minute before the interval when a corner from Trent Alexander-Arnold bounced to the back post and was nodded home by Fabinho.
The London side continued to prove resilient opponents after the break, although Diogo Jota struck the post in the 62nd minute after Alexander-Arnold had burst into the box, and Brentford goalkeeper Alvaro Fernandez did well to keep out another effort from the Portugal forward.
But Brentford stayed in the contest until the 69th minute when Oxlade-Chamberlain stooped at the back post to head in a wonderful cross from left back Andy Robertson.
Substitute Minamino wrapped up the win, tapping home after Brentford's attempt to play out from the back was intercepted by Roberto Firmino who set up the Japanese forward for a simple finish.
There was nearly a dream league debut for 17-year-old substitute Kaide Gordon, who was put through on goal but was denied by another good save from Fernandez.
"We needed these three points and a good performance so it was nice to do it in front of our fans. It will be good to grow our confidence," Fabinho said.
"The team played really well with the ball. The front three played really well. We created a lot of opportunities to score. It is a chance for everybody to show their quality and step up. We did that today."
Brentford remain in 14th place on 23 points.