Moore had a two-run double in the second inning, added a two-run home run in the fourth and had a solo shot in the sixth. The home runs were his first of the season.
The other Washington run came on an Ian Desmond homer in the eighth. He has nine home runs on the year.
The Nationals hit eight home runs in sweeping the three-game series in Toronto. Washington opened a 15-game string against American League East teams with series sweeps in Boston and against the Blue Jays. The six-game winning streak is a season high.
Strasburg (8-1) has also won six consecutive decisions. He allowed two runs and five hits over six innings. He walked one and struck out eight.
Toronto starter Kyle Drabek (4-7) allowed Moore's double and first home run in taking the loss. He walked two and struck out win while giving up seven hits in 4 1/3 innings.