It's been 40 since they rolled off five straight wins over the Rangers in one season.
Cleveland could change both Sunday by extending its longest home winning streak in two seasons to seven.
After Ryan Raburn delivered a walkoff three-run homer in Friday's 11-8, 11-inning victory, the Indians (55-48) only needed a leadoff first-inning homer by Michael Bourn off Yu Darvish to defeat Texas (56-48) again Saturday, 1-0.
"Everyone's pretty happy about (1-0 games)," said starter Justin Masterson, who combined with two relievers for a five-hitter. "... You feel like you're getting out there. Sometimes that helps keep that rhythm you have.''
The Indians, winners of three in a row to remain three back of AL Central-leading Detroit, are riding their best stretch in Cleveland since a 14-game run April 3-May 10, 2011.
Another victory would give Cleveland its first three-game home sweep in this series since Aug. 12-14, 1980. It would also mark the Indians' first 5-0 single-season run versus Texas since Aug. 10-25, 1973.
Cleveland's Ubaldo Jimenez (7-5, 4.49 ERA) gets the ball after suffering his first defeat in eight starts, 2-1 at Seattle on Monday when he surrendered two runs in 5 2-3 innings.
Jimenez, 3-0 with a 3.75 ERA over his previous seven outings, was tagged for two homers after giving up one in his last four starts combined. He also walked four in his 110-pitch outing.
"The main goal is to minimize the mistakes," Jimenez said.
The only significant error he made in a 5-2 victory at Texas on June 12 was serving up a solo homer to Leonys Martin. That was the only run, and one of four hits, Texas had off Jimenez.
The Rangers' offense needs to improve as the club is batting .240 with 2.6 runs per game during a 3-11 skid that's dropped them five back of Oakland for the AL West lead. Texas hasn't been this far off the division lead since finishing 10 back of the division-champion Los Angeles Angels in 2009.
''I know we can do better, but I'm not frustrated,'' manager Ron Washington said. ''When you get frustrated, you go blind and can't think, and I'm smart enough to know that.''
Rookie left fielder Engel Beltre had two hits Saturday in his second straight start, but Rangers manager Ron Washington said that David Murphy will return to the lineup Sunday. Murphy is 2 for 14 against the Indians this season, including 0 for 3 off Jimenez.
The Rangers will give the ball to Alexi Ogando (4-2, 3.13), who'll make his second start since missing more than a month with right shoulder inflammation. The right-hander went five innings in Tuesday's 5-4 loss to the New York Yankees, giving up three runs and six hits.
"(Ogando) gave us five innings of stressful pitching," Washington told the Rangers' official website.
Ogando, though, has been outstanding against the Indians, recording a 0.84 ERA in eight appearances. He threw five scoreless innings of relief versus Cleveland last season after posting a 0.64 ERA while winning both starts in this series in 2011.
The Indians have outscored the Rangers 22-12 since losing 6-3 in their first matchup this year June 10.