San Diego starter Clayton Richard delivered just two pitches before suffering an apparent left arm injury.
Stauffer (1-0) took over and threw 72 pitches over four innings to collect his first win in two years. He compiled a 9-12 record as a starter for the Padres in 2011 and has made only 15 major league relief appearances since then.
San Diego snapped a 1-1 tie with a three-run third inning, during which Chase Headley tripled home two runs and then scored on a single from Kyle Blanks.
Tyson Ross followed Stauffer to the mound and worked three innings. Dale Thayer gave up a run in the eighth and Huston Street turned in a perfect ninth for his 15th save.
Clayton Kershaw (5-5) was tagged with the loss, giving up four runs on seven hits in six innings.