The 39-year-old left-hander passed a physical and worked out with the Padres before their exhibition game against Seattle. He trained daily in nearby Tempe during the last two months while he waited for an offer.
"It's been a long offseason for me, frustrating at times, but I'm excited to be here, for sure," Thornton said.
If added to the Padres' 40-man roster, Thornton would get a $1.6 million, one-year contract and have the chance to earn $1.9 million in performance bonuses based on games pitched from 30-60.
"When you're older than the manager, sometimes that puts you in a bad position from other front offices," 38-year-old Padres manager Andy Green said, smiling. "But I'm glad to have him. He's going to be a nice piece for us."
The 6-foot-6 Thornton spent the past two seasons with Washington and had a 2.18 ERA in 60 appearances last season.
A 2010 All-Star with the Chicago White Sox, Thornton has also pitched for Seattle, Boston and the New York Yankees, mostly serving as a lefty specialist. He's made 60 or more appearances 10 times in his 12-year career.
Friedrich is joining his third team this winter. The Los Angeles Angels rescinded a waiver claim last month after he failed a physical and Colorado then released him.
Green said Friedrich is a "long-term play" as he recovers from a shoulder injury.