Manager Joe Girardi revealed the injury Wednesday after a 5-4, 11-inning loss to the Washington Nationals.
Miller told the Yankees about the injury Tuesday night but said he was feeling it for some time. He had an MRI exam Wednesday, which showed a slight strain in his flexor mass muscle.
Girardi said Miller will not throw for 10 days to two weeks.
"I think it was one of those things that was building," Miller said. "It hasn't been recovering quite like I wanted it to, so it's frustrating more than anything for me."
Miller is perfect in 17 save chances in his first season with the Yankees and has combined with Betances to form one of the most successful back-end duos in baseball.
Entering Wednesday, the Yankees were 28-1 when leading after seven innings and 32-0 after eight. Betances has allowed just one earned run in 32 1/3 innings this season and Miller has yielded three earned runs in 26 1/3 innings.