Scott Feldman pitched shutout ball for his last five innings, but fell to 0-5 in five starts since Aug. 5.
Texas manager Ron Washington was wary of playing Cleveland despite the Indians' poor second half.
His concerns were well-founded as the Indians, even with All-Star closer Chris Perez unavailable, ignored the Rangers' AL-best record and topped Texas 4-3 Saturday night.
"It's not always the best team that wins, but the team that plays the best," Washington said. "They're all major league teams and tonight they made the four runs they scored in the first stand up."
Cleveland broke its latest losing streak at six games as Jeanmar Gomez (5-7) and three relievers combined to give the Indians just their sixth win in 34 games since July 27.
Texas entered with a four-game lead in the AL West over Oakland and lost for just the fifth time in 16 games.
"You have to give credit to their bullpen," Washington said. "They certainly came in and did a good job. We were getting to Gomez there a little bit, but then their bullpen came in and really did the job."
Gomez gave up three runs and four hits over 5 2-3 innings. Joe Smith, Esmil Rogers and Vinnie Pestano finished the game. Pestano allowed two hits over 1 1-3 innings for his first save.
Pestano has been Perez's setup man for two years, but Cleveland's All-Star closer was never at the ballpark. He was with his wife Melanie and new baby daughter, delivered earlier in the day.
"The bullpen was terrific," Indians manager Manny Acta said. "To stop these guys is huge. At any moment (the Rangers) can turn around the score."
Cleveland scored four runs in the first inning off Scott Feldman (6-11) and improved to 42-24 when scoring four or more runs.
Carlos Santana had a two-run single before Casey Kotchman and Jack Hannahan each had RBI groundouts in the four-run rally.
"It was a nightmare first inning," Feldman said. "They came out swinging and it seemed like everything they swung at they got some hits. I dug a little too deep a hole with the way they were pitching."
Texas led the majors with 168 runs in August, and rallied to within a run.
David Murphy hit his 12th homer leading off the third for the Rangers' first hit. He walked to open the fifth and scored on a sacrifice fly by Mitch Moreland to make it 4-2.
Adrian Beltre's 26th homer with two outs in the sixth made it 4-3 and finished Gomez.
"I really wasn't concerned after the first inning because I figured at some point we would get back in it," Washington said. "But all we could scratch out was one run at a time. I thought we would put a crooked number up somewhere in the game, but we didn't."
Feldman pitched shutout ball for his last five innings, but fell to 0-5 in five starts since Aug. 5.
"He's not throwing that badly," Washington said. "There's one or two plays in an inning that causes his downfall."
Elvis Andrus hit safely against Cleveland for the 28th straight game, setting a Rangers record for a hitting streak against one team. Ivan Rodriguez had hit in 27 straight against Tampa Bay in 1998-2000.
Andrus extended the streak in his final at-bat in strange fashion. He took a big cut at a pitch from Rogers and sent a weak dribbler that hugged the third base line. It rolled foul at one point, weaved back into fair territory and Andrus laughed as he reached first base when the ball stopped at the third base bag.
Gomez started in place of Roberto Hernandez, out with a sprained right ankle. Hernandez left a start Monday because of the injury and tweaked it during fielding practice on Thursday. He will wear a protective boot for at least a week.
Cleveland also was without All-Star shortstop Asdrubal Cabrera, a late lineup scratch with a sore right wrist. Cabrera came in on defense in the ninth.
Notes: Russ Canzler, called up from Triple-A Columbus, went 2 for 4 as the DH in his Indians debut. ... Washington said he will try and get some playing time for prospect Jurickson Profar. When he does debut, the 19-year-old infielder will become the youngest Rangers player since C Ivan Rodriguez, also at age 19 in 1991. ... Washington wants to give Andrus some rest and used him as the DH. Andrus has missed only four games, starting 123 at shortstop. ... Beltre is batting .500 with seven homers and 15 RBIs in his past 10 games. ... Fans cheered as a member of the U.S. Navy Leapfrog team parachuted into the ballpark with the American flag during the national anthem.