Andrews steps up from being set-piece coach under predecessor Thomas Frank, now in charge of Tottenham.
Asked what fans can expect from his team, Andrews said: "I want a team that the fans feel represents them and what they want to see on a Saturday.
"We want to play winning football, we want to be competitive, we want to have an edge, we want to play dynamic, relentless football but we want to be organised.
“We want to have huge moments, huge games, and I think the big one is that we want to have an attitude and a relentlessness of progression. We really need to maintain that, and that’s on a daily basis.
“If you do that, the rest looks after itself.”
Massive advantage
On being promoted from within, Andrews is confident it will prove an advantage.
He explained: "It’s a massive advantage to know the players inside out, I’ve got a really good connection with the players. The ceiling is massive in terms of what we can achieve and I’ve got so much belief in them.
"I love being on the grass, I love developing young players, I love seeing the growth in young players as players and as human beings. There’s clear alignment between myself and the football club on that.
"There will be a clear plan to try and develop our talented young players and also our older players, because development doesn’t just stop when you get to a certain age; it’s for every single player, and we’ll support them in every single way we can."
Andrews also said directly to the Brentford support: “I’m humbled to be the new head coach of an amazing football club that’s touched me in a lot of ways.
"I love it here, I love coming in every single day, and I will give absolutely everything to continue the success of this football club."