The team announced that Saunders had been fired following a 103-99 loss to the New York Knicks on Sunday night, which dropped them to a league-worst record of 7-24.
Finch, 51, had success as a G-League head coach for the Rio Grande Valley Vipers and has interviewed for several head coaching jobs in recent years. He's spent time as an assistant with the Rockets, Pelicans and Nuggets before being hired by the Raptors in December.
Hiring a coach off another staff during the season is uncommon, but Toronto won't stand in the way of Finch's chance to become a head coach.
Finch and Timberwolves president of basketball operations Gersson Rosas worked together during their time with the Rockets.
Minnesota was hit hard with injuries and COVID-19 issues this season. Starting guard D'Angelo Russell played 20 games before recently undergoing left knee surgery that'll sideline him for the next four-to-six weeks. Star Karl-Anthony Towns has only played 11 games this season between a wrist injury and contracting COVID-19.
"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 was 43-95 as coach of the Timberwolves in two-plus seasons.
Minnesota travels to play Milwaukee on Tuesday night.