Mills was Terry Francona's bench for six seasons before the Astros hired him as manager last October.
"This probably should have happened a long time ago," Francona said of Mills' hiring. "But he looks very comfortable. He should. He's a good baseball guy. He's an even better person. He's organized. There's no reason he shouldn't be enjoying what he's doing. I think the Astros are fortunate. He's going to give them everything he has."
Oswalt struck out one and walked one. His previous turn was rained out, and he has pitched six scoreless innings in spring training.
"It was fun to watch him work because he worked quick," Mills said. "I think he felt really strong. He looks good out there, his demeanor on the mound is outstanding, and he's throwing the ball great. Any time you have Roy throwing the ball well, that's special because of who he is and the stuff that he has."
Jon Lester gave up one run and three hits, struck out two and walked two in his third start.
He threw 34 of 60 pitches for strikes. Lester said he worked on all of his pitches and is getting especially comfortable with his changeup.
"I threw a couple today behind and ahead in the count and was able to mix it up, not be predictable with it," Lester said. "I got some outs with it. I'm pretty comfortable with it right now and we just need to keep building that confidence with it because it is a feel and confidence pitch. You just keep building with it and keep throwing it."
Astros reliever Matt Lindstrom also threw a shutout inning and struck out one. He has not given up a run in any of his four outings this spring, a span of 4 2/3 innings.
"Physically I'm ready," Varitek said. "Catching-wise, I've got to get my catching joints in the right places because you can do all the drills you want, but it's a different monster once you get out there."
Varitek says the Astros players and the organization as a whole will be very pleased with their choice of leaders.
"We've been through a lot with Millsie as bench coach," Varitek said. "He's been a great support. It's good to see him getting the opportunity to run his own ship. The man knows the game. He's very organized in the things that he does. He's enthusiastic and he loves the game, hopefully that helps add to the winning tradition."
Francona said that he's proud of Mills being able to take the next step, but he says that his former bench partner is definitely in for some changes now that he's the head man.
"The hardest thing is, as a coach, when you go home, you sleep a little better," Francona said. "When you go home as a manager you don't. It doesn't mean you don't care, it's just harder to sleep."
Tuesday's game was the Astros' first sellout of the spring. ... Boston has a day off Thursday, its only one of the spring. ... Francona said that Josh Beckett would be at the Red Sox training facility in Fort Myers on Tuesday morning to try to get a side session in. Beckett was scratched from his last start on Sunday due to an illness. He was still feeling the effects of it yesterday, but Francona said he was confident that Beckett could get the work in Tuesday. If all goes well in the session, Beckett is expected to start on Friday.