The Rangers (82-66) brought a six-game winning streak into the series but lost 12-3 on Friday and then surrendered two runs in the eighth on Saturday before falling 2-1.
Texas trails the first-place Houston Astros by a half-game in the American League West but holds the second wild-card spot in the league.
On Saturday, the tying run scored on a two-out bloop single that landed just beyond the reach of diving left fielder Evan Carter, who made a similar highlight-reel catch earlier in the game.
"It's a game of inches; this is a tough one," Texas manager Bruce Bochy said. "No getting around it, but hey, come out here and be ready to go (Sunday)."
The Rangers averaged 7.8 runs per game during their six-game winning streak, but their bats have gone mostly quiet against Cleveland, especially in key situations.
Texas went 1-for-5 with runners in scoring position on Friday and 0-for-4 on Saturday.
"I wish I could explain it," Bochy said of the recent offensive struggles. "We'd fix it."
The Guardians (71-78) stranded seven runners through the first three innings on Saturday and were 0-for-8 with runners in scoring position when Tyler Freeman came to the plate with two outs in the eighth and runners on first and second.
Freeman, who had entered the game as a pinch hitter in the seventh, hit a two-strike blooper that fell just out of the reach of Carter and brought home the tying run.
"We've got to be prepared," Freeman said. "Whenever our name is called upon, we've got to be ready."
Steven Kwan followed with another two-strike, two-out single to score the go-ahead run.
Cleveland manager Terry Francona said it's comforting having Kwan at the plate in that situation. He said Kwan reminds him of Michael Brantley, who starred for Cleveland from 2009-18.
"It's just so nice when he's up," Francona said of Kwan. "He's going to give himself a chance to get a hit. More often than not, he'll hit the ball the other way, but that means he's not pulling off and he just gives himself a chance. That's a nice feeling when he's up there."
Another factor aiding Cleveland this season is defense. The Guardians are tied for the seventh-best fielding percentage in the majors (.987) and didn't commit an error for the second straight game on Saturday.
"We all know it's valuable," Kwan said of the defense. "We do our early work before the games. Even when people don't feel like it, we have people picking each other up, like, ‘Hey, let's get some reps.' Just 10 minutes, five minutes; it's accountability that helps people go."
The Rangers have not announced a starter for the series finale, while Cleveland plans to start rookie right-hander Gavin Williams (2-5, 3.43 ERA)
Williams did not get a decision in his most recent outing last Monday, when he allowed three runs and five hits in six innings of a 5-4, 10-inning loss to the San Francisco Giants.
Williams' only appearance against the Rangers came July 15 when he took the loss after allowing two runs and four hits in five innings.
—Field Level Media