CLEVELAND – For the second night in a row, the Twins eked out a bare minimum of offense, then watched a dramatic eighth-inning home run provide the game-winning cushion.

But there was a twist.

Instead of Griffin Jax surrendering the big blast, this time it was Guardians righthander Nick Sandlin who left a belt-high fastball over the middle of the plate to Willi Castro. The ball landed five rows deep in the right field seats, and the Twins won in the offensive desert known as Progressive Field for the first time this season, 4-1 over the Guardians.

And for Jax? This time, he saved the game, even getting the final out by himself when, with the bases loaded and the crowd of 17,391 roaring, Bo Naylor chopped a high bouncer up the first-base line.

"I was just ready to get the ball and get the game over with," Jax said after he fielded the ball and kept on running to first base. "I didn't want to leave it up to chance."

By now, the Twins know the drill when playing the Guardians, especially in Cleveland. They have met 11 times this year, and the Twins have scored a mere 32 runs in those games, more than four only once.

The solution? No choice but to make do with less.

"Everyone's going to have to elevate their play at this point to get where we want to be. We still control everything. We still control our own destiny," Twins manager Rocco Baldelli said. "To be at this point of the year, having struggled a little over the last month, and still be able to say 'Hey, if we go out there and play well from here on out, we're going to be pretty pleased with where we're sitting at the end of the season.' We just have to prove it."

The Twins achieved that commendable-if-normally-elusive goal on Tuesday, turning a pair of two-out RBI singles by Matt Wallner and Castro's first homer since Aug. 25 into all the offense they needed for the critical win.

"That was one of probably the most special ones this year," Castro said. "I know I've had a couple more games where I had some big hits, but today was really good, to get a two-run shot."

That shot allowed the slumping Twins, who have won only five of their last 14 games, to maintain their 1½-game lead over Detroit in the race for the final wild-card spot in the American League, and pull within 1½ games of Kansas City for their higher-seeded spot.

BOXSCORE: Twins 4, Cleveland 1

MLB standings

"That's what it's going to take if we're going to get to where we want to get to in the playoffs. Damn near every single game is going to be like that," said Wallner, who snapped an 0-for-17 run with his singles. "That's awesome to be part of now, and hopefully we keep doing that."

Jax was just happy to get a chance to redeem himself after Monday's big mistake.

"It's definitely a benefit, a perk of being in the bullpen. You've got to keep the [losses] to one night, [then] just eliminate what happened the night before," said Jax, who got the final two outs in relief of Jhoan Duran to earn his ninth save. "Especially this point in the year, you can't just sulk over what happened the day before."

Jax was the last of five Twins relievers to finish off Zebby Matthews' seventh career start. Matthews put runners on base in four of his five innings, but only once — when he centered a fastball in the strike zone for Guardians outfielder Lane Thomas in the fourth inning — did the rookie surrender a run.

Thomas drove the ball 423 feet to straightaway center, where it cleared the 19-foot wall, the ninth home run allowed by Matthews. But the Guardians never scored again.

"Zebby did a really good job. He faced a predominantly lefthanded lineup that the Guardians throw out there, and he pitched well," Baldelli said. "And then we turned it over to really a whole bunch of guys that all came in and pitched well."

Newcomer Cole Irvin retired Andrés Giménez on three pitches in his Twins' debut, and Cole Sands, Ronny Henriquez and Duran got through the sixth, seventh and eighth innings unscathed, Henriquez in part by picking pinch-runner Myles Straw off first base.

Duran was given a chance to record his first two-inning save in more than two years. But when he couldn't do it — José Ramirez led off the ninth with a double, and Thomas drew a one-out walk to bring the tying run to the plate — Jax was given a shot at redemption for Monday's loss.

Consider him redeemed.

"If we could have avoided Griffin and won the game, I would really have liked to have done so," Baldelli said. "But bringing him in to finish it off was the right thing."