Frank moved across London to join Tottenham Hotspur earlier this month, replacing Australian coach Ange Postecoglou, who had been sacked despite guiding the club to a Europa League triumph.
Andrews steps up after being hired as the set-piece coach under Frank and following roles with Sheffield United and Ireland.
"Keith is clear in what he wants, he's detailed, and the players and staff like and respect him," Brentford director of football Phil Giles said.
"He understands the players in the building and the way we've played and trained under Thomas Frank."
Brentford have become an established club in England's top flight since being promoted in 2021, producing upsets against some of the biggest teams.
But Andrews faces a challenge to preserve the club's place in the top division following Frank's departure and the sale of numerous first team players.
Goalkeeper Mark Flekken has left for Bayer Leverkusen and forward Bryan Mbeumo is a target for Manchester United. Midfielder Christian Norgaard is also a reported target for Arsenal.
"Keith will bring his own evolution but isn't starting from scratch, so hopefully we can make some quick progress," Giles said. "We were looking for somebody not just to maintain what we've been doing but actually try and get better."
6°C