Terms of the deal weren't immediately available.
Burnett, 30, has a 15-23 record with a 3.58 ERA in six seasons with Pittsburgh and Washington. He has logged 73, 69 and 70 appearances, respectively, with the Nationals in the past three seasons.
Burnett has held left-handed hitters to a .225 batting average and a .628 OPS during his career.
During the past week, Angels general manager Jerry Dipoto has added Burnett and Ryan Madson to his bullpen through free agency. Dipoto also traded reliever Jordan Walden to the Atlanta Braves for starting pitcher Tommy Hanson.