The boring way to characterize this game is that the difference in the final score was the difference in unearned runs.
The exciting way to characterize this game is that Andrew Benintendi would've delivered the game-tying double if anybody besides Byron Buxton were out there.
With runners on second and third and two outs in the ninth, Benintendi briefly appeared to triumph in his six-pitch battle against Jhoan Duran by smoking a 101.2 mph drive into the right-center gap. That vision dissipated as Buxton closed in on it, and he probably could've caught the ball without leaving his feet. As it stands, Buxton leapt for a final flourish and landed on the warning track with the ball secure in his glove, preserving Duran's save and a Minnesota victory in the series opener. The White Sox have lost 13 consecutive games at Target Field, and Buxton seems like he factors into every one.
The White Sox entered the ninth trailing 4-1, but applied immediate pressure when Lenyn Sosa reached on an infield single, and Brooks Baldwin drew his second walk of the season. Matt Thaiss pinch-hit for Bobby Dalbec and also walked, after which he was replaced by Michael A. Taylor on the basepaths.
The White Sox were 0-for-5 with runners in scoring position entering the inning, and it turned to 0-for-6 when Joshua Palacios struck out. Nick Maton made it 0-for-7, but his grounder to second scored Sosa to make it a two-run game, with the added bonus of advancing the other runners into scoring position. Were it not for Buxton, they would have crossed the plate.
But the White Sox made it harder on themselves than necessary with their defensive efforts earlier in the game. They outhit the Twins 10-5, but they out-errored them 2-0, and that didn't reflect all the flaws.
After the White Sox took a 1-0 lead on a Jake Amaya sac fly, the Twins tied it up. Davis Martin started his evening by walking Trevor Larnach, but Larnach should've been erased on Carlos Correa's grounder to first. Andrew Vaughn quickly fired an on-target throw to second, where Amaya was ready to receive it and step on the back, potentially in line for a return throw to first ...
... except Sosa cut in front of him. And that could've been fine, because Sosa was actually on the bag and the throw was within his reach, too. The problem was that Sosa didn't catch it. The ball instead clanked off his glove and rolled to the vacated shortstop position, and Larnach advanced to third, where he scored on Ty France's single to right. Ryan Jeffers then grounded into a double play to keep the first run unearned.
An inning later, Martin walked Luke Keaschall with two outs. Keaschall stole second, and Larnach followed with a dribbler up the first base line, where two issues arose. First, Edgar Quero's throw hit Larnach and caromed past Vaughn, and while Sosa was backing up Vaughn, Martin was not backing up Quero. Sosa fired to Quero because nobody was covering the plate, and Keaschall scored standing up.
Will Venable came out to protest the lack of an interference call, and was justified in doing so, because Larnach ran well inside the line:

But Martin locked up, and that didn't help matters.
The White Sox were also compromised by a looming bullpen game on Wednesday. Were Martín Pérez healthy and on schedule, Venable probably wouldn't have asked Mike Vasil to pitch a third inning to keep as many arms available over the remainder of the series. With many hands needed to make lighter work, Venable used Vasil for the final three innings after Martin, and Larnach stung him for a two-run shot before the night was through. There's another two-run difference right there.
Of course, the Sox offense could've simply scored two more runs itself, but numerous chances went by the boards. They had the bases loaded with one out in the first against Bailey Ober and while Amaya's sac fly was a decent contribution from somebody hitting .089, it turned out to be the peak of the first eight innings.
The next time Amaya came to the plate, he had runners on second and third after a two-out Dalbec double, but he popped out. An inning later, the Sox had runners on the corners with nobody out after singles by Nick Maton and Benintendi, but Luis Robert Jr. was caught looking at an outside-corner fastball, and Vaughn grounded into a double play. The Sox posted eight hits over six innings against Ober, but in the least helpful arrangement.
Bullet points:
*Robert came within a foot of homering off Ober in the first inning, with his line drive hitting off the very top of the left-field wall.
*Quero went 3-for-4 in a start against a righty, singling three times on Ober's changeup, and twice on 1-2 counts.