Trailing 2-1 entering their final at-bat, the Twins put together a walk and two hits for a pair of runs against the Tigers' bullpen, corralling their first victory of the season.
After Chris Parmalee flew out, Brian Dozier blooped a single, sending Carroll to third.
That brought up Escobar, who belted a pitch deep into center field that fell between outfielders Austin Jackson and Andy Dirks and bounced off the fence, allowing both runners to score for the walk-off win.
Glen Perkins (1-0) earned the victory with a scoreless top of the ninth for Minnesota.
Miguel Cabrera drove in both Detroit runs.