MLB.com ranks Odor as the Rangers' No. 2 prospect.
The 20-year-old was 1-for-2 on Sunday before exiting and is batting .290 on the season. He was replaced by utility infielder Luis Sardinas.
"He never slid," Rangers manager Ron Washington said. "He made a move to go back, and his shoulder came out -- never slid."
Odor's injury occurred just hours after the Rangers announced that first baseman Mitch Moreland would undergo left ankle reconstructive surgery and miss three months.
"Work as usual, come in prepared to play a ballgame," said Washington of his team's approach. "That's all you can do."
Grace Raynor is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.