Drew has been Atlanta's top assistant for the past six years. The terms of his contract weren't announced.
"After aggressively going through the interview process with all of the candidates it became clear, with his knowledge of the game as well as his experience as player and assistant coach, he had the qualities we were looking for to lead our club," said Hawks general manager and vice president Rick Sund.
Drew, also a former assistant with the Lakers, Pistons, Wizards and Nets, takes over a team that went 53-29 this season but were swept out of the playoffs by the Orlando Magic in the second round.
Woodson was fired after that exit, ending his Hawks career with a 206-286 record.