Tavarez had made an appearance in long relief on Sunday night after starter Tim Wakefield lasted just 2 2/3 innings. He was 0-1 with a 6.39 ERA.
"Last year, Julian saved us," Red Sox manager Terry Francona said. "He started for about three months and did a pretty good job. He has always been there for insurance. It was a real safety valve for me. You try to evaluate, 'Where do they come in handy, where are you using them?' We thought we can win more with the other pitchers. For as much as you use that safety value, it wasn't enough."
Tavarez had pitched only twice since April 24.
The decision to part with Tavarez impacted right-hander Craig Hansen, who had been bracing for a possible trip to the Minors. Hansen was recalled from Triple-A Pawtucket on May 5 and had allowed two hits and two runs in 2 2/3 innings.
Hansen said he first got wind of the move at lunch when he saw Tavarez saying goodbye to everyone.
"Julian is a great teammate," Hansen said. "I've learned a lot from him."
Hansen said he wouldn't have been downcast if he had been the odd man out.
"I would have gone down there and kept pitching," Hansen said. "It's all part of the game." While the 25-man roster is set, Francona continues to make out his lineup card with a mission of providing periodic breathers for the regulars while keeping others as sharp as possible.
Jacoby Ellsbury, who was hit on the knee by a pitch in Sunday's game, wasn't in the lineup for Monday's game against the Twins. Casey played first base with reigning AL Player of the Week Kevin Youkilis getting a rest. Alex Cora was again at shortstop as Julio Lugo recovers from a mild concussion.
Robert Falkoff is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.