Odor missed 12 games with a left knee sprain after a collision at home plate with Houston Astros catcher Martín Maldonado when trying to avoid a tag in a game May 4.
"I feel great, feel ready to go now," Odor said.
The Yankees also got good news regarding shortstop Gleyber Torres, who was cleared by the MLB/MLBPA joint committee after being placed on the COVID-19 injured list last Monday, according to manager Aaron Boone.
Torres, who tested positive for the coronavirus despite being fully vaccinated and previously having COVID-19, is on his way to Texas and is expected to be activated Wednesday, Boone said.
Odor, who batted fifth in the lineup, hit a single in his first at-bat in the second Tuesday night, then made a nifty pick up the middle to start an inning-ending double play in the bottom half of that frame. He also doubled in the fifth, finishing 2-for-5 on the night.
Odor, 27, played his first seven big league seasons for the Rangers, who designated him for assignment April 1 and left him off their Opening Day roster even when still owed two guaranteed seasons at $12 million each. He was facing Texas for the first time since then and said it was a weird feeling.
"It was tough. I think that's tough for every player, like you have to leave the team that you've been playing for all my career," he said about his discussion with Rangers officials at the end of spring training when they decided to let him go. "It's pretty tough, but you know what, that's already in the past. I'm here to play with this team, with my team now, with the Yankees. I'm just going to go there and compete and do the best I can to help the team win."
The Rangers are paying the bulk of the remaining portion of Odor's $49.5 million, six-year contract that also includes a $13.5 million team option for 2023 with a $3 million buyout. The Yankees were responsible only for a prorated share of the $570,500 minimum this year and the 2022 minimum, which is still to be negotiated. Texas would pay New York an additional $3 million if the option is declined.
Odor hit .164 with four homers and 11 RBIs in his 19 games with the Yankees before getting hurt.
In his seven seasons with the Rangers, Odor hit .237 with 146 homers and 458 RBIs. He hit .167 with 10 homers and 30 RBI in the pandemic-shortened 2020 season, then had a .200 average with two homers and five RBI in 35 spring training at-bats for the Rangers.
"He plays as hard as anybody, he's got a heart of gold and, you know, and it really was a tough decision to let him go," Rangers manager Chris Woodward said before Tuesday night's game. "But at the same time, I'm happy for him that he obviously caught on with these guys. Hopefully we can get him out a few times."
Woodward said he didn't even recognize the clean-shaven Odor when he saw him Monday. Odor maintained a shaggy beard while with the Rangers but had to shave it off when he joined the Yankees because of their team policy regarding facial hair.
Odor said his mother, who had always wanted him to shave the beard, was the one who cut it for him after he got traded. Odor said his former teammates, primarily Joey Gallo and Willie Calhoun, were giving him a good-natured hard time about his new look.
After Odor collided with Maldonado two weeks ago, he said he was initially concerned that the injury was much more serious.
"It was tough. Right away when that happened, it was really hurt," Odor said "I was scared. I never had problem with my knees. That play, I can't even get up. So I was a little bit scared. But thank God nothing serious happened, and I'm back. I'm ready to go."
ESPN's Marly Rivera and The Associated Press contributed to this report.