Towns grabbed an Andrew Wiggins rebound and dribbled once before draining the winning shot from the right side as the Timberwolves won their sixth consecutive game.
Towns celebrated by throwing his hands in the air as his teammates ran out onto the floor to mob him. Towns said the last shot was designed to go to Wiggins.
"Wiggins wanted the shot and we gave him the shot," Towns said. "I just wanted to be there for an offensive rebound in case he missed it. I am glad that when my team needed me the most, I made the shot."
Towns scored all 16 of his points in the second half and added 10 rebounds as the Timberwolves were behind twice in the extra session.
Jerryd Bayless had 19 points and a career-high 12 assists in the win.
Mike Conley had 26 points and eight assists for Memphis, which lost for the 12th time in 13 road games in front of the announced crowd of 13,000 at the Target Center.
Elsewhere, Jaylen Brown stepped in to fill some big shoes, finishing with 24 points and 10 rebounds as the Boston Celtics crushed the Charlotte Hornets 126-94 despite the absence of all-star Kyrie Irving.
Jayson Tatum also chipped in with Irving out with a hip injury, scoring 20 points, and Terry Rozier finished with a career-high 10 assists at the Boston Garden arena. Irving has missed the last two games.
Marcus Morris scored 15 points, Al Horford had 14 and Gordon Hayward 12 for the Celtics, who won for the seventh time in eight games.
Kemba Walker led the Hornets with 21 points, well shy of the 43 points he delivered in a 117-112 home win over the Celtics in November.
Malik Monk scored 16 points, Nicolas Batum added 13 and Marvin Williams 10 for the Hornets, who lost for the third time in five contests.
Morris drained a three pointer halfway through the third quarter to spark a 25-6 run by the Celtics.
Boston led 98-75 after three quarters and extended the lead to 35 points in the fourth.
Celtics coach Brad Stevens said he hopes to get Irving back for Friday's game against New York.
The Hornets played without Frenchman Tony Parker who sat out with a right ankle problem.
Also, Bradley Beal and Jeff Green combined for 48 points as the Washington Wizards used a balanced attack to beat the Indiana Pacers 107-89.
Beal scored 25, Green added 23 and the Wizards' reserves scored 58 points.
Thaddeus Young scored 13 points for the Pacers. They are 0-3 since a victory over Toronto in which leading scorer Victor Oladipo suffered a season-ending knee injury.
In the late game, Damian Lillard had 36 points and 11 assists to lead the Portland Trail Blazers to a 132-105 win over the Utah Jazz.
Guard CJ McCollum chipped in 30 points in the win while Donovan Mitchell had 22 points for the Jazz.