Odor had two doubles, and he knocked in his seventh and eighth runs in his last seven games. He's hitting .333 during that span.
"We have a pretty good offense," Odor said. "We can drive runs in. As soon as we score one run, we can score more."
Rangers ace Cole Hamels allowed three runs in 5 1/3 innings and left trailing by two runs, in danger of earning his second loss in 15 starts with the Rangers. But his teammates pounced on Baltimore ace Chris Tillman in the bottom half of the inning.
The Rangers sent nine men to the plate and scored five times on six hits. Mitch Moreland's run-scoring double and Ian Desmond's RBI single tied the game to set up Odor. Elvis Andrus added an RBI single to cap the rally.
"[Tillman] was making great pitches. He was trying the sinker, the curveball and changeup," Odor said.
The Rangers entered the series opener looking for their 13th consecutive victory with Hamels on the mound. The Texas lefty also entered with a personal nine-game winning streak, dating back to last season.
He took a no-decision in giving up three runs on seven hits and two walks and a home run. Hamels struck out four.
"We were able to get those big at-bats," Hamels said of the Rangers' sixth. "It's huge when guys are able to come through like they did."
Tom Wilhelmsen (1-0) earned the victory by recording the final two crucial outs in the sixth.
Relievers Keone Kela, Sam Dyson and closer Shawn Tolleson worked three scoreless innings for Texas (6-5), aided by a handy 4-6-3 double play started by Odor in the eighth.
Odor went to his right to make a back-handed spear on one knee of Matt Wieters' hard-hit ball. An overhand flip to Andrus at second started the inning-ending double play.