To make room for Stauffer on the 40-man roster, the team designated right-handed pitcher Thad Weber for assignment.
Stauffer, who started Opening Day for the Padres in 2011, had a 2-2 record with a 3.16 ERA over eight starts with Tucson.
"It was encouraging, from a health standpoint, it felt good and from a pitching standpoint, I was able to build on some things. I felt I was throwing the ball the way I needed to throw the ball," Stauffer said.
Stauffer signed a Minor League deal with the team before Spring Training after having surgery on Aug. 31 to repair the flexor tendon in his right elbow. The elbow felt good in Spring Training and then after the regular season started.
"There were times during Spring Training. But I think it wasn't until I was on a regular five-day starting [program] when I started feeling stronger and was able to go deeper into games," Stauffer said.
The Padres will carry, for now, three long relievers -- Anthony Bass, Tyson Ross and Stauffer.
"We'll see how it plays out … it's a nice luxury to have," said Padres manager Bud Black.
Stauffer had a June 1 opt-out clause in his contract, though he had no way of knowing if there were other opportunities with other teams.
"In that situation, you're showcasing yourself to other teams as well," Stauffer said. "Obviously, this was my first preference."
Stauffer, who was the fourth overall pick in the 2003 First-Year Player Draft by the Padres, is 23-31 with a 3.94 ERA in 96 games with the Padres, including 70 starts. His best season came in 2010 when he had a 1.85 ERA in 32 games.
Quentin, who missed Thursday's 6-2 loss to the Nationals, can miss a maximum of three games while on paternity leave. Quentin's wife, Jeane, gave birth earlier Thursday to the couple's first son, Clark.
Black said Quentin will return Saturday.
Weber had a 2.00 ERA in two games and nine innings with the Padres. He was 4-1 with a 3.93 ERA over six starts with Tucson.