Dango Ouattara needed just five minutes to put the Bees in front when he met Kristoffer Ajer's flick-on after Michael Kayode's long-throw wreaked havoc in the Liverpool box.
Kevin Schade added a second on the stroke of half time before Reds fullback Milos Kerkez pulled one back for his side just moments later.
An Igor Thiago penalty on the hour mark restored Brentford's two-goal cushion and while Mohamed Salah's 89th-minute effort threatened to set up a grandstand finish, Andrews' men hung on to condemn the visitors to a fourth successive Premier League defeat.
"We enjoyed it," Andrews told BBC Sport. "We have that type of approach to the game where we'll have a go and get into the opposition and have a good rattle, formulate a plan and see if we can execute it.
"That's underpinned by character, grit and togetherness and then the quality comes to the fore, and you can see that in the different types of goals we scored.
"I see things defensively in every team that I feel we can exploit because of the players we have. That's always the balance and it's only natural because they've lost a few games and the confidence might not be what it was at one stage and it's up to us to take advantage of that."
"It's not a given, far from it, because you're talking about serial winners and top Premier League international players. It was never going to be easy. We had to execute and make life uncomfortable for them and I felt we did that.
"I felt all week that there was an appetite to do something after Monday night's performance \[against West Ham United\] where we were brilliant and we carried that into \[Saturday\]."
Andrews was also quick to highlight the efforts of former Liverpool captain and Brentford midfielder Jordan Henderson, with last Saturday's victory the first time the Englishman has played against his former employers since ending a 12-year spell at Anfield in 2023.
"It was a really nice evening for him to showcase what he's still about," added Andrews.
"I spoke to him in the build-up about how proud he should be about what he's achieved as Liverpool captain.
"He's been a remarkable player for them and has proven to be a pretty good player for us."
Brentford's victory has lifted them up to 11th in the table, with the West Londoners next set to face giant-killers Grimsby Town in the EFL Cup on Tuesday evening.
That is followed by a return to Premier League action on Saturday when they visit Crystal Palace.