The Timberwolves trailed by 20 points in the third quarter before rallying to win game seven and the series 4-3.
Karl-Anthony Towns and Jaden McDaniels each scored 23 points for Minnesota, which will face the Dallas Mavericks in the Western Conference Finals.
The Indiana Pacers also advanced, defeating the New York Knicks 130-109 to win the Eastern Conference semi-finals 4-3.
Tyrese Haliburton scored 26 points for the visiting Pacers, who set an NBA playoff record by shooting 67.1% from the floor.
They will now face the Boston Celtics.
Minnesota won their series for the third time in Denver, and the result advanced them to the Western Conference finals for the first time in 20 years.
