Finch and president of basketball operations Gersson Rosas previously worked together with the Houston Rockets. Finch served as the head coach of the Rockets' G League affiliate when Rosas was an executive in Houston.
"I look forward to working hand and hand with Gersson to build and lead a team that Timberwolves fans will be proud of," Finch said. "We have excellent pieces in place and I can’t wait to get to work."
Finch was midway through his first season on Nick Nurse's staff. The former D-League Coach of the Year also had stints as an assistant with the Rockets, Denver Nuggets, and New Orleans Pelicans.
Minnesota didn't show much progress under Saunders' guidance, going 43-93 in parts of three seasons. The Timberwolves have lost eight of their past nine games and own the NBA's worst record at 7-24.
"We would like to thank Ryan for his time and commitment to the Timberwolves organization and wish him the best in the future," Rosas said in a statement. "These are difficult decisions to make. However, this change is in the best interest of the organization's short- and long-term goals."
Saunders, son of the late Timberwolves president and head coach Flip Saunders, took over for Tom Thibodeau midway through the 2018-19 campaign. He was the league's youngest head coach at the time of his hiring but had a decade's worth of experience as an assistant.