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."
Dellin Betances will move into the ninth-inning role.
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.
The San Diego Padres demoted former starting second baseman Jedd Gyorko to triple-A El Paso and recalled right-hander Nick Vincent. Gyorko was hitting .210 with two home runs and 11 runs batted in and lost his starting job to Cory Spangenberg. ... Tampa Bay Rays outfielder Desmond Jennings had arthroscopic surgery on his left knee. There is no timetable for his return. ... Seattle Mariners slugger Nelson Cruz, who leads the American League with 18 homers, was out of the starting lineup against the Cleveland Indians because of back spasms. ... Cincinnati Reds shortstop Zack Cozart hurt his right knee during the first inning of a game against the Philadelphia Phillies on Wednesday afternoon and had to be helped off the field.