A 3-pointer by Karl-Anthony Towns with 54 seconds remaining in regulation gave the T-Wolves a 129-126 lead, but James tied with a 3 with 45 seconds to go. James rebounded a Butler miss and the Cavs called time with 21 seconds to go.
James held the ball for the final shot and missed a 3 over Butler at the buzzer, sending the game to overtime.
The nip-and-tuck game became a 3-point shooting contest. The Cavs' biggest lead was eight points (119-111) on a Kyle Korver 3 with 6:10 to go.
The Cavs didn't have an answer for Towns' 3-point shooting as the 7-foot center made 6-of-6 from beyond the arc, improving his total to 87 this season.
Towns wasn't the only one who burned the Cavs from long range as the T-Wolves connected on 19-of-33, surpassing their season-high of 13. Butler made 4-for-5, Wiggins 4-for-8, Jamal Crawford 2-of-5 and Jeff Teague 2-of-4.
But the Cavs also showed consistent 3-point firepower, making 21-of-41, surpassing their season-high of 20. Smith led the way with 6-of-11, while Korver made 4-of-6, James was 5-of-7 and Thomas 3-of-5. Thomas, playing on his 29th birthday, had gone 1-for-16 from long range in the previous three games.
Recording 10 rebounds, James (5,907) passed Zydrunas Ilgauskas (5,904) to become the Cavs' all-time leader. James is just the third player in league history to lead a franchise in points, rebounds and assists, joining Michael Jordan of the Bulls and Kevin Garnett of the Timberwolves.
Also totaling 37 points and 15 assists, James recorded his ninth triple-double of the season and 64th of his career.
The Cavs have won five in a row at home over the Timberwolves. Since 2014-15, the Cavs are 7-1l versus the T-Wolves.
With Dwyane Wade resting on the second night of a back-to-back, Lue finally went to rookie Osman to provide some energy. Osman showed plenty of hustle on defense during his 22 minutes, which the Cavs had forgotten how to do of late.
While the Cavs had struggled to play a 48-minute game for months, they were engaged from the start, taking a 31-29 lead at the end of the first quarter and trailing 66-64 at the half. They went ahead 101-99 going into the final period with James scoring 13 points in the third quarter.
The Timberwolves' Wiggins, the first overall pick by the Cavs in 2014, became the sixth youngest in league history to score 6,000 career points. He reached the milestone with a 3-pointer with 9:28 to go in the second quarter. James was the youngest at 21 years, 89 days. Wiggins is 22 years, 349 days old.
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