Turner on Thursday left the Dodgers' 7-2 win over the Miami Marlins after suffering the injury while trying to score in the seventh inning.
Turner had an MRI for the injury Friday, after which the Dodgers announced the move.
"We're just going to play it by ear, and see how it goes from day to day," Turner said. "We'll make sure we get it to 100 percent before we try to do anything else."
Turner is hitting an MLB-best .379, with a home run and 17 RBIs.
Dodgers manager Dave Roberts said he was hopeful Turner will be back "in a couple of weeks."
"It's a big blow," Roberts said. "All the components to help you win baseball games are what J.T. does."
The Dodgers recalled outfielder Brett Eibner from Triple-A Oklahoma City to complete their roster.
Turner pulled up after rounding third base and hobbled toward home plate, only to be tagged out after the throw from the outfield.
"My second or third step past third base, it kind of grabbed on me, almost like a cramp. In hindsight, I should have stopped right then and there. Tried to make my way home. I don't think it had any effect on the injury," Turner said.
On Friday night, outfielder Yasiel Puig left the game in the fourth inning of Dodgers' win over Miami with lower-mid back tightness.
Puig appeared to injure his back on a checked swing that resulted in a walk. Roberts said he wasn't sure when Puig would be back in the lineup, but Puig said he hopes to play Saturday or Sunday.
"I'm not going on the DL," Puig said, according to the Los Angeles Times. "I need to be back on the field, to keep hitting, and go to Miami for the All-Star Game. I'm not going on the DL."