The 2-0 shutout victory in the second of a three-game set snapped a five-game losing streak for the Tigers, who are third in the American League wild-card standings.
Detroit (62-54) had also lost six in a row to the Rangers dating to last season.
"Whenever you've had a string of losses like we had earlier in the season or we had these past five days, it definitely feels better to get the monkey off your back," Tigers manager Brad Ausmus said.
The Tigers can return to Comerica Park for a seven-game homestand starting Monday with momentum back on their side by capturing the series finale on Sunday afternoon in what promises to be a sun-scorched Globe Life Park.
Getting their first win the series was a start.
"At the end of the day, it's just one game," Tigers fill-in third baseman Casey McGehee said. "We're trying to make a playoff push. Every game is important."
McGehee was a big part of Saturday's win, even though he wasn't expected to be a big part of the club this season. The 33-year-old veteran was signed by Detroit to a minor-league contract in February, released the next month and signed again.
After the injury to Nick Castellanos earlier this month, McGehee was called up. He has hit safely in five of six games since being recalled, capped by a 4-4 performance in the 2-0 win. The four hits tied a career high.
"I wouldn't have continued to play if I didn't think I could do it," McGehee said.
Ausmus has not been surprised by McGehee's contributions since joining the team.
"I was never concerned," Ausmus said. "Casey McGehee looks like a big league player."
The Tigers are sending rookie Michael Fulmer (9-3, 2.43 ERA) to the hill. The right-hander is making his 19th start of the season and gunning for his fourth consecutive quality start.
He did lose his last outing on Aug. 8 at Seattle after allowing two runs and five hits in seven innings. The 23-year-old rookie of the year candidate is facing Texas for the first time.
The Rangers counter with A.J. Griffin (5-1, 4.38) in the rubber game. The 28-year-old veteran is hoping to fare well against the predominately right-hand-hitting Tigers. Griffin has held right-handed batters to a .194 (27-139) average this season, the fifth lowest in the American League.
He does have a 6.39 ERA in this last six starts beginning July 20 but is 2-1 in that span. The Rangers have won his last four starts. Griffin took a no-decision on Aug. 9 at Colorado, a 7-5 come-from-behind Texas win. He is 1-1 in three career starts against the Tigers.
Despite scoring only two wins, the Tigers did rack up 13 hits Saturday against Cole Hamels and 14 total.
"There's potential danger up and down their lineup," Texas manager Jeff Banister said.