After dominating the opening set, the third-seeded Swiss battled back to force a tiebreak and then broke the Italian in the final game of the match to finish a 6-1 7-6(4) 7-5 victory.
"I was trying to be a little more aggressive," said Wawrinka, who blasted 57 winners, including 26 off his forehand, in the Louis Armstrong Stadium contest.
"I'm very happy the way I won the match."
Wawrinka will next face 64th-ranked Daniel Evans of Britain, a 6-4 6-4 5-7 6-2 winner against Germany's 27th seed Alexander Zverev.
Wawrinka, who has emerged from the shadow of compatriot Roger Federer after winning the 2014 Australian Open and 2015 French Open, was asked if he thought he was a threat to win the title at Flushing Meadows.
"I'm number three in the world with a great career so far," the 31-year-old Swiss said.
"I'm happy with what I'm doing so far in my career. I'm happy the way I'm playing so far in this tournament. Let's see what can happen."
Wawrinka said he was pleased with the competitive level of the match, especially given they had to deal with a rain delay after starting and liked the pace on the Louis Armstrong court.
"I think in general was a great level," he said. "I'm feeling well, playing some good shots.
"Armstrong I think is a little bit faster than what it used to be, a little bit faster than the other courts."
(Editing by Greg Stutchbury)
