Harper's blast came in the fourth inning after Washington had also scored twice in each of the preceding two frames, grabbing a 6-1 advantage.
Denard Span contributed two hits with an RBI and made a pair of sparkling catches in center field for the Nationals, who notched their third straight over the Reds.
Dan Haren (2-3) tossed six strong innings and also knocked in a run en route to the victory. The right-hander yielded a pair of runs on six hits, didn't issue a walk and struck out five.
Rafael Soriano hurled a scoreless ninth for his seventh save.
Reds starter Mike Leake (1-1) was touched for four runs (three earned) on six hits over three full frames.
Brandon Phillips went 3-for-4 with an RBI for Cincinnati.