He steps up from his role as set-piece coach following Thomas Frank’s departure to Tottenham.
Andrews, who was assistant manager of his country under Stephen Kenny, has signed a three-year contract.
His first Premier League game in charge will be at Nottingham Forest in August.
Lewis Mason is at Brentford’s training ground and says Andrews is following the same path as predecessor Thomas Frank.
Tune in to the news at the top of the hour for the latest reaction from Brentford’s training ground.