CLEVELAND -- Griffin Jax has openly acknowledged that the Twins’ pitching staff has been “running on fumes,” and manager Rocco Baldelli noted before the game that the Twins simply had to go out there and “prove it” if they truly wanted to control their destiny and hold this playoff spot.

On Tuesday, they dug deep and cobbled together a way.

It took two more players busting out of slumps -- Matt Wallner with a pair of RBI singles and Willi Castro with a clutch two-run homer -- and a pitching plan seemingly held together by wishes and duct tape, but the Twins scratched and clawed their way to a 4-1 victory over the Guardians at Progressive Field on Tuesday to temporarily stop the bleeding.

“I think that's what it's going to take if we're going to get to where we want to get to in the playoffs,” Wallner said. “Damn near every single game is going to be like that.”

The win ensured that the Twins’ lead over the Tigers in the Wild Card race stayed at 1 1/2 games with 11 to play for Minnesota, with Detroit winning in extra innings in Kansas City later Tuesday evening. The Twins own the tiebreaker over the Tigers.

While Castro played the late hero with his two-run homer in the eighth for his first extra-base hit since Aug. 27, the Twins’ depleted bullpen still had to piece together a high-wire act with unexpected contributions to get through the game, especially in the wake of Jorge Alcala’s surprise option to Triple-A after Monday's game.

“The more we tried to map it out, the more we stopped trying to map it out,” Baldelli said. “Because we realized we would probably have to treat it inning by inning and decide it. When Zebby [Matthews] came out of the game, we kind of did it all on the fly.

“We just had to see where we stood, see how far each of these guys could go.”

That involved Cole Irvin -- ostensibly brought in to give the Twins length -- for one lefty-on-lefty out to get the final out of Matthews’ fifth inning, with Cole Sands then bridging the gap to Ronny Henriquez, who provided an unexpected high-leverage seventh inning in a one-run game, involving a huge pickoff of pinch-runner Myles Straw as the tying run.

Henriquez had pitched to a 2.60 ERA in 17 1/3 innings this season entering Tuesday, nearly all low-leverage -- but given the state of the Twins’ bullpen, he very well may need to keep stepping up into those spots.

“He doesn’t have just one strength where you’re trying to find a spot in the lineup for him,” Baldelli said. “He can be that guy that pitches in those other spots where maybe you don’t have someone particularly to face three or four hitters. He can do it because he has almost a starter’s arsenal.”

Jhoan Duran emerged in the eighth for his first multiple-inning appearance since June 14, then Jax entered amid the ninth for redemption after having allowed the game-winning homer on Monday to cap a high-leverage pitching sequence that would have been difficult to imagine even a month ago.

Castro helped give the Twins the cushion they needed by crushing a Nick Sandlin fastball a Statcast-projected 371 feet to right field in the eighth, continuing the Twins’ trend of late-innings run-scoring this season, while Carlos Santana built on his Gold Glove defensive résumé with a diving stop in the bottom of the eighth.

“I think 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, I think that was really good, that inning, to get a two-run shot.”

The Twins still couldn’t breathe easily until the very end, as Duran left two runners on base in the ninth before Jax emerged and allowed a two-out infield single to load the bases, bringing the winning run to the plate.

It took one pitch to Bo Naylor for Jax to get the game-ending groundout and for the Twins to finally breathe easy, six pitchers into their makeshift plan.

That’s what it looks like for the remaining pieces of this roster to “prove it,” it seems.

“When some other guys stay healthy for us, it’s a different scenario, but we’ve got what we’ve got, and thankfully the guys that keep getting the call are stepping up when we need them to most,” Jax said.