The Rangers put Griffin on the 10-day disabled list and recalled right-hander Dillon Gee from Triple-A Round Rock. Griffin will undergo an MRI when the Rangers return home Monday.
"It got pretty tight there [against] that last batter," Griffin said. "It felt more like a cramp earlier on in the game. Sometimes those come, and you can just pitch through them. It just kept tightening more and more, grabbing more and more."
One batter before he exited, Griffin allowed a grand slam to Toronto second baseman Devon Travis.
Griffin (4-2) allowed three runs and four hits in 1⅓ innings. It was the Rangers' shortest start of the season.
The Associated Press contributed to this report.