Rodriguez was hit on the right hand by a line drive hit by Detroit's Ramon Santiago with one out in the seventh inning with the ball caroming off Rodriguez's hand to first baseman Kendry Morales for an unusual out.
"I just tried to get the ball and it bit me," said Rodriguez with a laugh.
Rodriguez, though, was evaluated by team doctor Lewis Yocum on Tuesday and X-rays found no fractures in his hand.
"It's sore, but it doesn't seem swollen too bad," Scioscia said. "There's no fracture, so we'll take it day-to-day and see where he is."
Rodriguez, who had his hand wrapped in a bandage, said his plan is just to rest for a few days before trying to play catch.
"I'm going to take it easy today," Rodriguez said. "I'll be all right maybe in a couple days."
Rodriguez, 24, has a 6.57 ERA this season in 14 appearances with the Angels. Monday was his first appearance since being recalled from Triple-A Salt Lake on Sunday in place of right-hander Sean O'Sullivan, who was optioned to Salt Lake.
Rhett Bollinger is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.