Washington remained four games in front of second-place Atlanta in the National League East with six games remaining. The Nationals also stayed a game ahead of Cincinnati for the best record in the majors.
The Nationals gave up three runs in the first inning and Gio Gonzalez (21-9) shut down the Phillies after that.
Gonzalez, baseball's winningest pitcher this season, gave up six hits and struck out six over six innings.
Morse hit a solo homer in the second to bring the Nationals within one and his ground out in the fourth brought home a run that forged a 3-3 tie. His second home run was a two-run shot in the sixth.
Tyler Cloyd (2-2) allowed six runs on six hits in five-plus innings to take the loss.
The game was Philadelphia's last at home this season. The Phillies produced a 40-41 record in their own ballpark after having a winning mark at home during each of the last 11 seasons.