At 1-1, it's the first time the Twins have been .500 since 2010, as they got off to bad starts in both '11 and '12 and never recovered.
"It was a heck of way to win a ballgame," Twins manager Ron Gardenhire said. "You just have to keep playing, and that's what we did today. It's a big boost out there to get a win under your belt against a great team over there. That was fun baseball."
Escobar entered the game after Wilkin Ramirez served as a pinch-hitter for shortstop Pedro Florimon in the seventh inning. Ramirez plated the Twins' first run with an RBI double off reliever Brayan Villarreal.
Escobar made his second at-bat of the season a memorable one, as he lifted the ball to deep left field, and it kept carrying before dropping in near the wall between Andy Dirks and Austin Jackson for a two-run double.
"It was a fastball right over the middle," Escobar said, with Ramirez acting as his translator. "It was just my first swing, and I got it right down the middle on the first pitch. I thought I did my job, because I hit it in the air and hit it far enough, but I didn't even think it was going to go that far and bring the other run in."
Dozier, who was at first base after singling to right field on a 1-2 fastball that advanced Carroll to third with one out, made the proper read on the fly ball and was able to beat the throw home to score the winning run.
"The ball carried pretty good, but off the bat, it sounded like he hit it pretty good," Coke said. "I was watching it, but with the shadow changing and the sun, I couldn't tell exactly what the ball did. I thought it bounced in the bullpen. I thought I gave up a homer."
But the walk-off hit wouldn't have been possible without a solid Minnesota debut from right-hander Kevin Correia.
Correia, who signed a two-year deal worth $10 million this offseason to join the Twins, gave up two runs on seven hits -- all singles -- and a walk over seven strong innings.
"It was fun," Correia said. "I wasn't really nervous about it all, but it's a beautiful park, and that's the first time I've gotten to pitch here in my career. So I was pretty comfortable immediately. I wasn't trying to get a feel for any of the park or anything. It's a good pitcher's park, and I was comfortable on the mound from pitch number one."
Tigers third baseman Miguel Cabrera, last year's American League MVP, brought home both runs for the Tigers against Correia.
The first run came in the third inning on an RBI single after Jackson walked with two outs and Torii Hunter singled. The hit went under the glove of Trevor Plouffe at third and a diving Florimon at shortstop.
Cabrera came through again with another two-out RBI single in the fifth, when he laced a ball to right field to score Omar Infante, who singled and reached third on a single from Hunter. But Hunter was thrown out by right fielder Chris Parmelee while trying to advance to third to end the inning.
"That guy's a tough out," Correia said. "He's the best hitter in the world. He didn't hit the ball hard off of me. He's just so good. That's what he's trying to do is hit it where he hit it and get those RBIs."
The Twins' bullpen also came through, as Jared Burton and closer Glen Perkins each tossed a perfect inning in relief of Correia.
It set the stage for Minnesota's comeback in the ninth, which was the club's first walk-off win since Sept. 13 of last season against the Royals.
"It's a new year," Dozier said. "It's just our second game in. Losing on Opening Day was a tough one, but at the same time it was good to get in the win column, and now we have the chance to win the series tomorrow."