Anderson, who came to Boston in a July 28 trade with the White Sox for Mark Kotsay, was the only player Boston non-tendered.
In 86 games split between Sox -- White and Red -- Anderson hit .243 with four homers and 18 RBIs in 2009. He played 19 games for the Sox.
The Red Sox were able to agree to terms on a one-year deal with Boof Bonser, the right-hander they acquired in a trade with the Twins on Thursday.
The 28-year-old Bonser didn't pitch in 2009 after undergoing arthroscopic surgery in February to repair partial tears to his labrum and rotator cuff. He has made 96 appearances in the Major Leagues, going 18-26 with a 5.12 ERA.
Bonser gives the Red Sox depth in their rotation should one of the regular starters sustain an injury.
To complete the trade for Bonser, the Red Sox also announced Saturday night that they have sent Minor League right-hander Chris Province to the Twins.
Province, 24, posted a 2-4 record with one save and a 2.60 ERA in 43 relief appearances at Double-A Portland in 2009.
Among the players the Red Sox tendered contract offers to prior to the deadline were closer Jonathan Papelbon, setup men Manny Delcarmen, Hideki Okajima and Ramon Ramirez, as well as first baseman Casey Kotchman and outfielder Jeremy Hermida.
Those players are eligible to file for arbitration between Jan. 5-15.
Ian Browne is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.