Takahashi appeared in 53 games this year with the New York Mets in his first season in the majors. He went 10-6 with eight saves and a 3.61 ERA.
"We believe that his contributions in various roles in our bullpen will be extremely important to the overall success of our club," Angels general manager Tony Reagins said in a release.
The 35-year-old Takahashi made 12 starts and became the first Mets pitcher since Roger McDowell in 1986 to record double digits in victories and eight or more saves in the same season.
Takahashi agreed to a minor league deal with the Mets in February after 10 seasons with the Yomiuri Giants in Japan's Central League.
The Angels also have approached agent Scott Boras about closer Rafael Soriano, so shoring up the bullpen appears to be a major focus.
Also Thursday, the Angels did not tender a contract to infielder Kevin Frandsen, making him a free agent. Frandsen batted .250 with 14 RBIs in 54 games with Los Angeles last season.
Information from ESPNLosAngeles.com's Mark Saxon and The Associated Press contributed to this report.