Both Bryant and Garnett watched from the bench as the young core for both teams went back and forth during the final minutes.
Los Angeles' Russell scored 13 points in the fourth quarter alone, including a runner in the lane with two seconds remaining in regulation to tie the game at 114.
Guard Andrew Wiggins missed a turnaround jumper at the horn on the other end but scored four of his team's nine points in overtime as Minnesota snapped a four-game losing streak and finished a four-game homestand 1-3.
Towns finished with 26 points and 14 rebounds for his 10th double-double in 21 games this season.
"It was great, for both of us, it was great (to see the young players step up)," Bryant said of himself and Garnett. "It seems like yesterday, we were the young ones. It's just crazy lining up with KG after all these years. It's nuts; where did time go?"
Wolves guard Kevin Martin led all scorers with 37 points on 14-of-27 from the floor, including 6-of-9 from beyond the arc. He scored 17 of those points in the fourth quarter, making four 3-point field goals, helping the Wolves erase a five-point deficit with under 10 minutes to play.
"We were moving the ball around tonight and just taking what the defense was giving us," Martin said. "Ricky (Rubio) was running the show, he was calling a lot of plays for me, so that makes it easy. It was just fun to get a 'W.'"
The night was especially good for Martin considering he appeared to injure his shooting wrist on Minnesota's first possession of the game. His performance was reminiscent of his 37-point performance against the New York Knicks on Nov. 19, 2014, when it was revealed the next day that he broke his wrist early in the game.
Martin said he will get X-rays on his right wrist Thursday.
"I saw him when he dinged his wrist, but I didn't look at him because he was making shots," Wolves interim coach Sam Mitchell said. "The last thing I wanted to do was ask if he was OK. He was going to have to find me."
Towns was consistently good, too, scoring nine points in the first quarter and leading all players with 15 points and eight rebounds at the half.
With Los Angeles down by three headed to the fourth quarter, it was Russell who shone brightest for the Lakers. He buried back-to-back 3-pointers to pull the Lakers within two with 51 seconds remaining. At one point, he scored nine consecutive Los Angeles points.
His long 2-pointer earlier in the quarter gave the Lakers their largest lead of the night at 99-94 with 7:26 to play.
"I thought he played well," Lakers coach Byron Scott said. "He was doing some of the things that we talked about doing. Just looking to attack and be aggressive, making plays for himself and his teammates."
The Wolves rallied from there. Martin drilled a pair of threes sandwiched around two free throws from forward Gorgui Dieng. His final triple of the quarter gave the Wolves a 102-99 lead, and they never trailed again.
Wiggins finished with 19 points, and forward Shabazz Muhammad had 15 points and five rebounds off the Minnesota bench. Rubio had 12 assists, the sixth time this season he reached double digits.
Bryant finished with 11 points on 5-of-13 shooting. His 1,513 career points against the Wolves are second only to former teammate Karl Malone's 1,677.
Garnett had six points and four rebounds and is just two defensive rebounds shy of Malone's NBA record of 11,406.
NOTES: Wednesday marked the 78th game between Minnesota F Kevin Garnett and Los Angeles F/G Kobe Bryant. ... Timberwolves rookie PG Tyus Jones scored 27 points in his NBA Development League debut Wednesday for the Idaho Stampede. Selected 24th by Cleveland before being traded to Minnesota on draft night, Jones had one point and one assist in two games for the Timberwolves this season. ... The Lakers will play the seventh game of an eight-game road trip against the Spurs in San Antonio on Friday. ... The Timberwolves open a two-game road trip the same night against the Nuggets in Denver.