He had left Saturday's 3-1 win over the Boston Red Sox due to lower back pain after throwing just seven pitches. The 30-year-old Mills (0-1, 9.68 ERA) struck out Jarren Duran and left after allowing a double to Rafael Devers.
Leiter (2-2) picked up the win after allowing three hits, striking out five and walking one in 5⅓ innings.
"It was nice to feel that emotion," said Leiter, who exited to an ovation from the crowd. "Proud of being able to kind of hold us where we were and not have to run through other guys and really use a lot of arms."
Mills was activated June 7 from the 60-day injured list for a right quadriceps strain. He said the pain he felt Saturday is "a very similar feeling" to his previous injury, and he will see how he recovers.
"The pitch to Duran on the strikeout was when I initially felt a little bit of something," Mills said Saturday. "It didn't get any better, thought maybe I could get through it."
Right-hander Michael Rucker was recalled from Triple-A Iowa on Sunday in a corresponding move.
The Associated Press contributed to this report.