The two best postseason games the Twins have ever won are probably the two you would expect: Game 6 and Game 7 of the astounding 1991 World Series. Both games check all of the boxes you could hope for in a best postseason win - hugely important, team won the World Series, legendary plays, heroic individual performances (or sports-heroic, at least).
So naturally, I am picking... neither of them. Instead, I'm going to cheat just slightly and pick a game that's not technically from the postseason - but it may as well have been, and in terms of in-game excitement, it easily surpasses both of the 1991 contests.
It's the 2009 AL Central one-game playoff: Twins 6, Tigers 5 (12). Both teams had to play all-out through the scheduled end of the regular season, so neither necessarily had the starter it wanted available for this game; however, the Twins were still able to use Scott Baker, who was roughly as good as ostensible ace Nick Blackburn, while the Tigers had a much larger dropoff from Justin Verlander to Rick Porcello.
The only hitter on either team to reach in the first inning was AL MVP Joe Mauer, who doubled with two outs and was left on. Detroit threatened in the top of the second, as Miguel Cabrera led off with a double and Ryan Raburn singled him to third with one out, but Baker escaped. The Tigers capitalized much more effectively on their rally in the third; Curtis Granderson walked with one out, moved to second on a grounder, and scored the game's first run on a Magglio Ordonez single. Cabrera then homered to make it a 3-run lead.
Minnesota got one run back in the bottom of the inning. Matt Tolbert led off with a single, Denard Span singled him to second, and Orlando Cabrera flied out to move him to third. With a 2-0 count on Mauer, Porcello attempted to pick Span off of first and threw the ball away, allowing Tolbert to score. Mauer then walked, putting the tying run on base, but Porcello recovered to strike out Jason Kubel and end the inning.
The fourth and fifth innings passed quietly; the Tigers put a runner on in each one, but one was erased on a double play and the other was stranded at first. The Twins, meanwhile, were baserunner-free until there were two outs in the bottom of the sixth, at which point Kubel homered to halve the deficit. Porcello then walked Michael Cuddyer and was pulled in favor of Zach Miner. Miner allowed a single to Delmon Young and hit Brendan Harris with a pitch to load the bases before Tolbert flied out to end the inning.
Detroit threatened to extend its lead in the top of the seventh, as Brandon Inge's leadoff walk chased Baker from the game; a trio of relievers worked the remainder of the frame, with Matt Guerrier eventually inducing a forceout from Placido Polanco to leave runners at the corners. In the bottom of the inning, Minnesota took its first lead of the day when Nick Punto singled and Orlando Cabrera homered. However, the Tigers countered immediately, as Ordonez led off the top of the eighth with a longball that tied the game at 4. Guerrier then issued a pair of walks, but Joe Nathan relieved him and left both runners on.
Brandon Lyon was spotless in the bottom of the eighth, and Nathan allowed singles to Ramon Santiago and Granderson in the top of the ninth, but recovered to keep them from scoring - thanks in part to Ordonez lining into a double play. Lyon and Fernando Rodney combined to work around a pair of walks (one of them, to Mauer, was intentional) in the home ninth and sent the game to extras.
Jesse Crain took over for Nathan in the top of the tenth and appeared to be on his way to a simple inning, with a hit batter sandwiched between two outs. Inge then doubled, scoring pinch runner Don Kelly with the go-ahead run. The Twins countered quickly against Rodney, as Cuddyer led off with a triple. Young grounded out and Harris walked; Tolbert then singled Cuddyer home and moved pinch runner Alexi Casilla to third with the winning run. Punto followed with a lineout to left, and Ryan Raburn cut Casilla down at home in a failed attempt to score the winning run.
Crain, Ron Mahay, and Bobby Keppel combined to retire the Tigers in order in the eleventh, and Rodney stayed in the game for the bottom of the inning and matched the trio of Twins. Keppel had a much tougher time of things in the top of the twelfth, starting with a one-out walk to Miguel Cabrera. Don Kelly singled Cabrera to third, and took second on the throw. Raburn was then intentionally walked to load the bases, still with one out, bringing Inge to the plate. Keppel's first pitch to Inge may or may not have hit him; the umpire ruled that it didn't, and Inge went on to ground into a force at home. Gerald Laird then struck out to extinguish the threat.
Rodney took the mound once more in the bottom of the twelfth. Carlos Gomez led off the inning with a single, moved to second on a groundout, and scored the game-winner on a Casilla single.
This is obviously a terrific game; Sports Illustrated selected it as the regular season game of the decade. It has a few lead changes in regulation, followed by a comeback in extra innings, followed by the winning run getting thrown out at the plate, followed by the bases loading with no runs scoring, followed by the actual walkoff hit. The game amassed a WPL of 7.72, which is the fifth-highest ever among postseason games (if you count it as a postseason game). That's without accounting for the situation (winner take all), or the excellent performances from great players (Mauer reached base four times, Miguel Cabrera had a two-run homer early), or the maybe-HBP that would have given the Tigers the lead in the twelfth, or the fact that the pinch runner who was thrown out at home in the tenth went on to drive in the winning run two innings later.
You can obviously make an entirely valid case for selecting either of the last two games from the 1991 World Series; those games would dominate the remainder of the discussion so thoroughly that there's no point in even listing an Honorable Mention section down here. There is one more consideration with respect to the Twins, however; they also played a couple of outstanding World Series before moving to Minnesota, and those should be taken into account.
Which they will be - next time, when we cover the best postseason victory in the history of the Washington Senators.