Garlick entered the game in right field in the seventh as a replacement for Max Kepler, who hit his 13th career home run at Progressive Field in the fourth, also a three-run shot.
Cleveland tied the game in the ninth when Jordan Luplow, replacing injured Franmil Reyes in the cleanup spot, brought home the tying run with a squeeze bunt.
The Twins responded quickly against Karinchak (2-1). Luis Arraez was the automatic runner at second base, and Josh Donaldson walked. Garlick homered to center field on a 3-2 pitch.
Not only did the home run give Minnesota its first extra-inning win in nine tries, it also earned Garlick a Gatorade shower from Kepler.
''That was my first time,'' Garlick said of the celebration in front of Minnesota's dugout. ''It felt really good.''
Garlick's postgame interview with reporters was interrupted by a hug from teammate Miguel Sano. The Twins took two of three from their AL Central rivals after arriving in Cleveland at 4:30 a.m. on Friday following a doubleheader at the Los Angeles Angels on Thursday.
Karinchak had allowed one run and three hits with 38 strikeouts in 20 2/3 innings going into the game.
''The kid's human,'' Indians manager Terry Francona said. ''That's the (fourth) hit he's given up of the year. That's pretty amazing.''
Hanser Robles (1-1) walked Cesar Hernandez to start the ninth, and Amed Rosario reached on an infield single for his fourth hit of the day. Hernandez went to third when Jose Ramirez flied out to deep center. Luplow laid down a bunt on the first pitch and Hernandez slid home.
Francona said Luplow bunted on his own. Eddie Rosario struck out and Owen Miller, playing in his first major league game, flied out.
Tyler Duffey pitched the 10th for his first save.
The extra-inning loss capped a rough day for the Indians, who announced before the game that Reyes will miss five to seven weeks with an internal oblique strain. Reyes leads the team in RBIs and is second in home runs.
Cleveland starter Zach Plesac retired his first nine hitters, but the Twins broke loose in the fourth for the second time in the series after trailing 3-0.
Arraez and Donaldson started the inning with singles. Kepler, playing his 40th game in Cleveland, sent a 2-0 pitch down the right-field line to erase the deficit. He has 17 career homers against the Indians overall.
Rob Refsnyder's single gave Minnesota the lead. Andrelton Simmons' RBI groundout ended the day for Plesac, who was charged with five runs - three earned - in 3 2/3 innings.
The Twins scored nine times in the fourth inning of Friday night's 10-0 win.
Minnesota starter J.A. Happ allowed four runs and struck out a season-high 10 in six innings.
Kepler, who has been playing with a sore hamstring, also singled in the fifth. Minnesota manager Rocco Baldelli said the Twins were monitoring Kepler throughout the game and decided to pull him for Garlick in the seventh.
A few innings later, Garlick got the biggest hit of the day.
''Sometimes things do play out like that,'' Baldelli said. ''Garlick hit it to center field. Karnichak is about as nasty at it gets.''
BIG LEAGUE DEBUT
Miller, who batted .406 at Triple-A Columbus, was called up to take Reyes' roster spot. He started at designated hitter and was 0 for 5 with three strikeouts.
Miller got the news following Saturday night's game in Toledo. The call-up led to some unexpected change in family plans since Miller's brother, Noah, was scheduled to attend his graduation ceremony at Ozaukee High School in Wisconsin on Sunday.
''They ended up skipping it,'' Owen Miller said. ''They said that getting to play in the big leagues for the first time is a little more rare than graduating high school. My brother was perfectly fine with missing that to make the trip to Cleveland.''
Twins: RHP Kenta Maeda is on the 10-day injured list with a right adductor strain. He was removed after five innings Saturday. ... DH Nelson Cruz (bruised left wrist) and INF Jorge Polanco (ankle) remain sidelined.
Indians: C Roberto Perez, who had surgery for a broken right thumb this month, was transferred to the 60-day injured list.
Twins: RHP Matt Shoemaker (2-5) will start the opener of a three-game home series Monday against Baltimore.
Indians: LHP Sam Hentges (1-1) will start the first of a four-game series in Detroit on Monday.
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