Taj Gibson tallied a game-high 28 points and Butler added 24 for the Timberwolves, who completed a three-game homestand with two victories.
Julius Randle had a team-high 23 points for the Lakers, who went winless on a three-game trip to Dallas, New Orleans and Minnesota.
Down 91-84 entering the final period, the Timberwolves got a pair of 3-pointers from super-sub Jamal Crawford and one from fellow reserve Tyus Jones in a 10-4 burst that made it a one-point game with 9:35 to play.
The teams then traded leads six times, with two ties interspersed, before Butler gave Minnesota the lead for good at 105-103 with three free throws at the 5:25 mark.
Gibson's three-point play with 3:59 to go gave the Timberwolves some breathing room, and the Northwest Division leader held on from there to beat the Lakers for the third straight time this season.
Gibson hit 11 of his 16 shots from the floor, while Butler went 10-for-11 at the line for Minnesota, which outscored Los Angeles 29-18 on free throws.
Jeff Teague had 20 points, Crawford 15 points and Karl-Anthony Towns 10 points and a game-high 19 rebounds for Minnesota.
Randle hit eight of his 15 shots for the Lakers, who were outshot 46.7 percent to 45.7 percent by the Timberwolves.
Ivica Zubac backed Randle with 19 points, while Kentavious Caldwell-Pope and Brandon Ingram added 17 apiece.
Zubac and Caldwell-Pope both completed double-doubles with a team-high 11 rebounds. Caldwell-Pope also matched Teague for game-high assist honors with six.
Isaiah Thomas, in just his third game for the Lakers since being acquired from Cleveland last week, shot just 3-for-15 and finished with seven points.
The Lakers led for nearly the entire first 42 minutes of the game, but every time they appeared to be taking control, the Timberwolves rallied.
Los Angeles went up by as many as 15 points in the second quarter, only to see Minnesota close within 65-62 by halftime.
And after the Lakers re-established a 12-point lead in the third, the Timberwolves scratched, getting within 91-84 headed into the fourth period and setting up their 35-20 fourth-quarter finish.
--Field Level Media