The world number two is now unbeaten in 16 hard court matches after overcoming Harrison 6-4 6-2 6-2 in two hours and eight minutes.
Nadal, seeded second for the competition, is one of the favourites to win in New York, having tasted success in both Montreal and Cincinnati following his shock first-round exit at Wimbledon.
The Spaniard – who has won 12 grand slam titles, including the 2010 US Open – faced stubborn resistance from Harrison in the first set with the world number 97.
However, Nadal stepped up his game in the remainder of the match – breaking Harrison twice in the second and third sets to seal a comfortable win.
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