Hudson (3-1) tamed the Nationals on one run over seven innings, yielding just three hits while walking two and striking out six in breaking a two-start winless streak.
With the Braves leading 2-0 in the second, Hudson led off with a double and scored on Chris Johnson's single.
He socked his third career homer in the fifth, extending Atlanta's lead to 6-1.
Freddie Freeman went 3-for-5 with three RBI and Evan Gattis knocked in a pair for the Braves, who took their second straight against the Nats.
Washington starter Gio Gonzalez (2-2) struck out nine but walked five over four full innings, yielding five runs on seven hits.
Wilson Ramos drove in the only run for the Nationals, who dropped a third consecutive game overall.