Jaden McDaniels scored a career playoff-high of 30 points and Anthony Edwards added 29 for the Timberwolves, who closed the game on a 13-1 scoring run.
With Slovenian star Luka Doncic slowed by illness, James led the Lakers with 38 points - surpassing the playoff record for a player past his 40th birthday, the 32 points scored by Kareem Abdul-Jabbar in the NBA Finals in 1987.
The teams had split the first two games of the best-of-seven series in Los Angeles. The Timberwolves host game four on Sunday looking to take a 3-1 lead.
James added 10 rebounds, four assists, two steals and two blocked shots, drilling a trio of three-pointers in the fourth quarter that helped the Lakers keep it close.
They tied it at 103-103 with 4:37 remaining on a basket by Doncic, who was battling a stomach ailment that Lakers coach JJ Redick said had him throwing up throughout the day.
Edwards responded with a three-pointer and Naz Reid followed with another. After driving for a layup Edwards blocked Rui Hachimura at the other end and after McDaniels pushed the lead to 116-104 with 39 seconds left Redick pulled his starters.
Magic, Bucks claw back
The Orlando Magic held off the defending champion Boston Celtics to cut the deficit in their playoff series to 2-1 as the Milwaukee Bucks clawed back a game against the Indiana Pacers.
The Bucks, who, like the Magic, had dropped the first two games of their series on the road, also turned the tables upon returning home for game three with a victory over the Pacers.
The victories were crucial, with the Magic and Bucks both aware that no NBA team has come back from a 3-0 series deficit to win a playoff series.
Germany's Franz Wagner scored 32 points and handed out eight assists and Paolo Banchero added 29 points with six rebounds for the Magic in Orlando - where the Celtics erased a 12-point fourth-quarter deficit but were thwarted in the frantic final minutes.
Six-time All-Star Jayson Tatum scored 36 points for the Celtics, showing no sign of the wrist injury that caused him to miss game two -- his first missed postseason game since he came into the league in 2017.
Boston's Derrick White drove for a finger roll layup to tie it at 91-91 with 2:31 remaining.
But Wagner made back-to-back baskets to put Orlando back in front, 95-91 with 1:26 to play.
Boston pulled within two on White's dunk with 28.1 seconds left but Orlando held on, the Celtics unable to score on a final inbounds attempt with three-tenths of a second left as the game ended 95-93.
"High intensity," Banchero said of the final few possessions. "I love moments like these."
With Banchero leading the way, the Magic erased an early eight-point deficit to lead 31-27 after the opening period.
But Boston used a 14-0 scoring run in the second quarter to regain the lead, Tatum delivering eight straight points in the spurt and Jaylen Brown capping the first half with a three-pointer that pushed Boston's lead to 59-49 at the break.
The Celtics couldn't press the advantage; however, the Magic produced a run of their own in a third quarter in which they out-scored the Celtics 24-11.
Boston had 21 turnovers, leading to 26 Orlando points, and they connected on just nine of their 27 three-point attempts.
"It was just a great team effort," Banchero said. "We knew it was going to be a hell of a game at home. We wanted to get the win, have a chance to tie it up 2-2. We're excited for the opportunity."
Bucks 117-101 Pacers
In Milwaukee, two-time NBA Most Valuable Player Giannis Antetokounmpo and Gary Trent Jr. scored 37 points apiece for the Bucks.
Trent drilled nine three-pointers on 12 attempts - equalling the Bucks' record for treys in a playoff game set by Ray Allen.
"It's a testament to my hard work, a testament to my teammates finding me," Trent said. "It's a blessing - we needed this win to continue to keep going."
Trent's performance helped the Bucks rally from a 10-point halftime deficit. They out-scored the Pacers 39-18 in the third quarter to take control, leading by as many as 20 in the fourth.
Pascal Siakam led the Pacers with 28 points while Tyrese Haliburton had 14 points and 10 assists.
Both the Bucks and Magic will try to level their series on Sunday.