Belt scored four times and his blast in the fifth inning handed a 6-0 lead to San Francisco, which opened a homestand by snapping a three-game skid.
Vogelsong (2-4) allowed just three hits over five shutout innings, walking one and striking out two in posting his first victory since April 11.
But he exited the game after being hit in the right (pitching) hand with a pitch while batting in the sixth inning. The Giants said Vogelsong will be out six weeks with a fractured right hand.
Washington starter Zach Duke (0-1), filling in for the injured Ross Detwiler, was roughed up for four runs on seven hits over 3 1/3 innings as the Nationals stumbled to a third straight loss.