X-rays revealed the extent of the injury. Smith ran the bases and then was replaced by Jeff Francoeur in the field to start the bottom of the seventh inning.
The Braves said Smith will travel on Sunday night to Atlanta, where he will be further evaluated on Monday.
Smith was 0-for-3 with a run scored Sunday in the Braves' 6-0 victory over the Mets. Julio Teheran pitched a one-hit shutout, allowing just a Michael Conforto single to lead off the third inning, as the Braves won their fifth straight game.
Smith, a 23-year-old rookie, is hitting .241 with three home runs and 21 RBIs in 166 at-bats this season. He leads the Braves with 14 stolen bases.