With the Nationals tailing 1-0 entering the frame, pitcher Jordan Zimmerman and Steve Lombardozzi led off with singles and Harper followed by slamming the first pitch from Roy Halladay (4-4) into the right-center field gap for two RBI.
He came home one out later on Adam LaRoche's sacrifice fly and Desmond followed with a solo shot as Washington took control of the game.
Zimmerman (3-4) held the Phillies to one run on seven hits over six innings, walking two and striking out two for the Nationals, who won their sixth straight in Philadelphia and logged their 13th win in 16 meetings with the Phils overall.
Halladay sustained his first career loss against the Nationals franchise, falling to 11-1 lifetime against them. The Phillies ace was roughed up for five runs on nine hits over six innings.
Erik Kratz hit his first career homer for Philadelphia, which fell for the fourth straight game.