Since then, Dempster has steadied himself and given Texas the sort of production it hoped for when it pulled off the July 31 deal. The right-hander will take the mound against the Indians on Wednesday at Rangers Ballpark riding a four-start winning streak, during which his ERA is 1.73.
"I've been pleased with my ability to stay focused on the task at hand and not look back," Dempster said. "Keep looking forward. I've been pretty much doing the same thing I did with the Cubs, I just had a couple of starts early where I didn't execute a few pitches."
Texas won Tuesday's opener after beginning the day with a three-game lead over the A's in the American League West. The club also has adjusted its rotation to give its starters extra rest, meaning Dempster has enjoyed six days off since beating the Royals on Wednesday.
The 35-year-old Dempster has made 23 starts and pitched 147 innings this season. He was on the disabled list twice while with Chicago.
"It's always nice to get extra rest, but I've missed six starts this year already, so it's not as big of an issue," Dempster said. "I'm used to being at about 170-180 innings this time of year. My arm feels good, but it's always nice to get a couple extra days."
The Indians, who have lost four straight to continue their second-half slide, will hand the ball to Jeanmar Gomez.
The 24-year-old right-hander has surrendered 12 home runs in 82 2/3 innings, with a 44-to-31 strikeout-to-walk ratio. He has put up a pair of solid performances against Texas, going 2-0 and giving up six runs in 12 2/3 innings.
Gomez's last outing, Friday at Minnesota, was his second-shortest of the season. He threw 59 pitches and walked three while giving up four runs in three innings.
"Gomez just wasn't aggressive enough," manager Manny Acta said afterward of his decision to make the early pitching change. "He didn't throw even 50 percent first-pitch strikes and he had already given up four runs."
Indians: Ailing Cabrera could miss rest of trip
The good news for shortstop Asdrubal Cabrera is that his right wrist injury is not believed serious enough to end his season. The bad news is that he might miss the rest of Cleveland's series at Texas, which ends on Thursday.
Cabrera, who has dealt with the problem off and on all month, sat out his second straight game on Tuesday. Acta said the All-Star could be ready to come off the bench for defensive purposes on Wednesday and hopes to have him back in the lineup on Thursday.
General manager Chris Antonetti reaffirmed his confidence in Acta's abilities on Tuesday, when he made a planned visit to Rangers Ballpark.
Rangers: Oswalt dealing with forearm injury
Pitcher Roy Oswalt was diagnosed by Dr. Keith Meister on Tuesday as having a strained right forearm. He will not throw for two to three days and then be re-evaluated by Meister.
He left Sunday's start against the Rays after two innings when the injury flared up on him. Martin Perez took over and pitched five innings in long relief. That leaves the Rangers without both of their long relievers at least through Wednesday.
Mike Napoli was 0-for-2 and was hit by a pitch in a rehab game for Double-A Frisco on Tuesday night. He caught seven innings and is scheduled to catch nine on Wednesday as he returns from a quad injury.
Tuesday's win gave the Rangers a 4-3 edge in the season series.
Dempster gave up an unearned run on two hits over six innings on Aug. 31 at Cleveland, striking out seven.