Stauffer (1-3), who has received little run support this season, logged his first win in 11 starts, holding the Nationals to four hits while yielding only a seventh-inning solo homer to Laynce Nix.
The right-hander surrendered four walks and struck out four.
Heath Bell nailed down the win with his 11th save in the ninth inning for San Diego, which evened its three-game series with Washington at one win apiece.
Rookie outfielder Blake Tekotte notched his first two major league hits, including an RBI triple, and Kyle Phillips also knocked in a run for the Padres.
Jordan Zimmermann (2-6) absorbed the hard-luck loss, surrendering a pair of runs on five hits over six innings, walking one and fanning four.