Dodgers manager Don Mattingly said Puig got hit "near the top or on the side a little bit."
Puig, considered day to day, was not in the lineup for Sunday's game.
"As the game went on, he got tight," Mattingly said, according to MLB.com. "We knew he wasn't going to hit an inning before, but we knew he could play defense."
Puig's hand was swollen, but X-rays came back negative, MLB.com reported.
Cardinals outfielder Matt Holliday also left the game early, departing in the fifth, four innings after he was shaken up while diving back into second base on a pickoff attempt.
The Cardinals said team doctors determined Holliday did not sustain a concussion and he is in Sunday's lineup batting in the third spot.
The Associated Press contributed to this report.