Arcia gave the Braves a 7-6 lead in the 10th inning when he drove in automatic runner Eddie Rosario.
But Arcia hurt himself coming around first and limped into second before pulling up lame and being tagged out as he overran the bag.
Braves manager Brian Snitker said the diagnosis is a left hamstring injury that will land Arcia on the IL.
"He heard it pop, and that's never good," Snitker said.
The Red Sox tied the game in the bottom of the 10th, before Austin Riley had a go-ahead, two-run single in the 11th inning to rally the Braves to a 9-7 win.
Arcia, 28, is hitting .243 with seven home runs and 26 RBIs this season.
The Associated Press contributed to this report.