The left-handed Vargas got an $8.5 million deal, while right-handed Hanson will make $3,725,000. Both veterans are expected to be key components of the Angels' rotation this season.
Los Angeles acquired Hanson from Atlanta on Nov. 30 in a deal for reliever Jordan Walden. Hanson, who grew up in Southern California as an Angels fan, made $535,000 last year.
Hanson went 13-10 last season in his fourth straight year of double-digit victories for the Braves, although his ERA rose to a career-worst 4.48.
The Angels then picked up Vargas in a deal with Seattle for Kendrys Morales on Dec. 19. Vargas, who made $4.85 million with the Mariners last season, also is a Southern California native who pitched at Long Beach State.
Vargas went 14-11 for the Mariners last season with a career-best 3.85 ERA and 141 strikeouts. He has started 96 games in the past three years for Seattle.
The trades overhauled the Angels' rotation after the departures of Dan Haren and Ervin Santana. Los Angeles also added right-hander Joe Blanton to the rotation behind ace Jered Weaver and C.J. Wilson.
Reliever Kevin Jepsen and third baseman Alberto Callaspo are the Angels' only remaining arbitration-eligible players.
Jepsen, who made $501,000 last season while going 3-2 with a 3.02 ERA in 49 appearances, asked for $1.4 million and was offered $975,000.
The Angels are expected to close a deal soon with Callaspo, their regular third baseman for most of the past two seasons. Callaspo, who made $3.15 million last season while batting .252 with 10 homers and 53 RBIs, asked for $4.7 million and was offered $3.65 million.