Mitre entered spring training with an outside chance to win a spot in New York's starting rotation, as well as an opportunity to pitch long relief. But Ivan Nova appears to have won the No. 4 spot, and Freddy Garcia is the leading candidate for the fifth spot, with Bartolo Colon also in the mix.
The Yankees also signed free agent Kevin Millwood to a minor league deal Friday.
Mitre's departure clears the way for both Garcia and Colon to make the team, although it is unclear who will get the starting job and who will be assigned to the bullpen for long relief/spot starting duty, which was Mitre's job with the Yankees in 2010.
"We talked about three of the four guys going [with the team for opening day] and now you take one away," Yankees manager Joe Girardi said. "Could things change? Yes, they could. But we've said all along that it would be, probably, three out of the four."
Mitre, 30, was 0-3 with one save and a 3.33 ERA last season. After a good start this spring, he pitched poorly in his last appearance, allowing two home runs to the Orioles in Sarasota on Tuesday.
The Brewers have been scrambling for another arm because of injuries to their starting rotation. Zack Greinke will begin the season on the disabled list because of a rib injury suffered in a pickup basketball game, and Shaun Marcum has been bothered by tightness in his shoulder this spring.
"Sergio, he's going to be in a good spot to, obviously, help Milwaukee with their recent rash of injuries," Yankees general manager Brian Cashman said.
Mitre is 13-29 with a 5.27 ERA in seven seasons with the Cubs, Marlins and Yankees. He signed a $900,000 deal with New York in early December.
Dickerson, 28, broke into the big leagues with Cincinnati. He joined Milwaukee in August in a trade for Jim Edmonds.
Dickerson gives the Yankees backup insurance if center fielder Curtis Granderson, out with a strained muscle on his right side, is not ready for opening day next Thursday.
"It gives us another choice as an alternative if Curtis doesn't make opening day," Cashman said. "We're not going to treat opening day as a pressure point. I don't think it's a long-term issue, but I can't tell you whether it's a few days or a week or 10 days."
Dickerson is batting .278 in 17 games with Milwaukee this spring. A Yankees source said he believed Dickerson would be assigned to Triple-A Scranton.
Senior writer Jerry Crasnick covers Major League Baseball for ESPN.com. Information from ESPNNewYork.com's Wallace Matthews and The Associated Press contributed to this report.