The veteran will look to win his fifth consecutive start when he faces the reeling Cleveland Indians on Wednesday night in Arlington.
Dempster (5-1, 4.36 ERA), acquired in a trade with the Chicago Cubs on July 31, began his Rangers tenure by allowing 19 runs over his first three starts - far below expectations after posting a 2.25 ERA with the Cubs.
He seems to have settled in now, though, winning his last four outings with a 1.73 ERA.
The right-hander gave up two runs and struck out eight - his most in seven starts with Texas - in six innings of a 7-6 win over Kansas City last Wednesday.
''You know, I was in a situation where I was pitching for pride (with the Cubs),'' Dempster said. ''Now I'm just trying to do my best every fifth day.''
Dempster allowed an unearned run and two hits while striking out seven in six innings of a 5-3 win over Cleveland on Aug. 31, improving to 2-0 with a 0.98 ERA in three career starts versus the Indians.
AL-best Texas (84-57) beat Cleveland 6-4 in the series opener Tuesday in its return home from a 10-game road trip, which began by taking two of three against the Indians from Aug. 31-Sept. 2.
Adrian Beltre, hitting .383 with six home runs over his last 11 games, hit his 31st homer and is 11 for 22 with four homers versus Cleveland in 2012. His home run Tuesday hit the top of the wall and was initially ruled a double. The umpires ruled it a homer after reviewing the replay.
''The way it bounced ... it had to hit something,'' manager Ron Washington said.
Indians manager Manny Acta disagreed.
''In the replays we have here it looks kind of inconclusive," he said. "Talking to the umpire, he felt it hit some metal bar back there behind the padding.''
The Rangers have won 12 of their last 16 at home and 11 of the last 13 meetings in Arlington with the Indians (59-83), who are tied with Minnesota for the AL's worst record.
Cleveland has lost 22 of its last 26 on the road, and general manager Chris Antonetti, who has previously stated his confidence in Acta, is in Texas to observe the series.
''Things haven't gone right the whole second half," Acta said. "We're battling, we're keeping our head up and trying to finish strong."
The Indians will look to end a five-game road losing streak to the Rangers with help from Jeanmar Gomez, who has been solid versus Texas this season.
Gomez (5-7, 5.33) beat the Rangers on Sept. 1 after giving up three runs - including solo homers by Beltre and David Murphy - in 5 2-3 innings of a 4-3 victory for his second win versus Texas in 2012. He gave up three runs in seven innings of a 6-3 victory May 4.
The right-hander, though, wasn't as good during Cleveland's 7-6 win over Minnesota on Friday, allowing four runs and walking three in three innings.
"Gomez just wasn't aggressive enough," Acta told the team's official website.
Josh Hamilton and Ian Kinsler are both 0 for 6 off Gomez.
- Cleveland Indians
- Ryan Dempster