Washington trailed 5-2 heading into the bottom of the second inning, but scored eight of the game's last nine runs.
Belliard's two-run homer in the sixth started Washington's comeback and Orr's two-run seventh-inning single broke a 6-6 tie. Alberto Gonzalez and Austin Kearns had three hits apiece and Emilio Bonifacio added two hits in the Nationals' second straight win following a nine-game losing streak.
The winner was reliever Charlie Manning (1-2).
Cincinnati scored five times in the second on Joey Votto's three-run double and Jay Bruce's two-run homer.
Paul Bako had one RBI and Bruce had two hits for the Reds, who have lost seven of eight.
Losing pitcher Mike Lincoln (1-3) allowed four runs -- three earned -- and three hits in 1/3 of an inning.