CHICAGO -- White Sox acting manager Miguel Cairo stood in the Conference and Learning Center at Guaranteed Rate Field following the Guardians' 10-7, 11-inning victory over his squad Tuesday night and took the blame for this highly disappointing setback.

But the White Sox players weren’t going to let Cairo stand alone following this direct hit to any remaining hope they have to capture the American League Central.

“We had a bunch of stuff,” said White Sox left fielder AJ Pollock, whose 11th inning home run marked the 1,000th hit of his career. “A couple missed opportunities throughout the game. A couple defensive miscues. It was execution on our end for sure. Miggy, I thought he did a good job of getting the guys out there and putting guys in the right spots. We just didn't win."

The White Sox (76-72) entered this three-game series against the American League Central leaders knowing they needed a sweep to have a true chance at making the playoffs for a third straight season. Sure, they espoused the 'one game at a time' theory, which has worked for them over the past three weeks, but the math dictated the above needed outcome.

Instead, their extra-innings setback not only dropped them five games back of the Guardians (81-67), but also left them at 7-10 in head-to-head competition. With the Guardians owning the tiebreaker and no Game 163s on the docket this season, the deficit is more like six games.

So, has hope been completely dashed? The White Sox still have Lance Lynn and Johnny Cueto taking the mound for the next two days, albeit against the Guardians’ best in Triston McKenzie and Shane Bieber.

“Just not giving up. Bringing out our best tomorrow and taking it day by day,” said White Sox starter Dylan Cease, who allowed one run over six innings. “The external situations are going to be what they are. We just have to focus on what we can do.”

“I've been in situations where the math doesn't seem right,” Pollock said. “The key is to just win that next game. The team above you feels that. If we can get them to feel that, it's a fundamentally sound ballclub, but let's see what they feel like when you get a little closer and feel us. That's the goal.”

This series opener was a hard-fought contest on both sides, with players giving that playoff-like effort in September. White Sox second baseman Josh Harrison went full dive into right field to take a hit away from Amed Rosario in the fifth, after he already robbed Rosario of an infield hit with another great play two innings earlier.

There wasn’t anything specific Cairo would share about the disappointment in his job done, aside from the team losing. Maybe the acting manager was trying to take a little pressure off his players with their collective backs against the wall.

“I should have done a better job today. I let my team down,” Cairo said. “I have to come back tomorrow and show them, you know, just another day and we have to perform. I have to perform, too … We lost the game, so it’s on me.”

As Pollock explained, there were enough issues all around to plague the White Sox. Jimmy Lambert, who has been strong in relief all season, walked the first two hitters faced in the seventh in protecting a 3-1 lead, and the Guardians eventually tied the score.

Even after the White Sox scored two off usually impervious Cleveland closer Emmanuel Clase in the 10th, they couldn’t finish off the victory and were touched up for five in the 11th off Jake Diekman. Cleveland swiped three bases in the 11th, with Rosario scoring on the final steal of third as catcher Seby Zavala’s throw went past Yoán Moncada, who wasn’t covering at the base.

It was a rough finish to an exciting game. The Guardians knocked the South Siders’ elimination number down to 10, but of equal importance, they now own the tiebreaker.

“That’s awesome, but anything can happen,” said Cleveland center fielder Myles Straw, who delivered the game-winning two-run double in the 11th. “It’s baseball. I’ve seen crazier things happen. Just trying to take these one day at a time and try to go win tomorrow.”

“We've got to win the series, got to win tomorrow in order to have a chance of that,” Pollock said. “And just keep fighting."