Murray, the 2012 champion and playing just his fifth tournament of the year after undergoing hip surgery in January, last featured in a major at Wimbledon in 2017.
Murray, whose world ranking has slumped to 382, goes on to face Fernando Verdasco, the Spanish 31st seed after seeing off the error-prone Duckworth, ranked at 448.
“He was serving big in the first set and the ball was flying around,” said 31-year-old Murray who only returned to the tour in June.
“But I started to get into the return games, stood back a little more and that gave me more time to get in the rallies and find my range with my ground strokes.”
Like Murray, former champion Wawrinka also sat out the 2017 US Open — in his case because of two knee surgeries.
But he showed no sign of rustiness on Monday, defeating eighth-seeded Grigor Dimitrov of Bulgaria for the second successive Grand Slam.
Wawrinka, the 2016 champion, swept to a 6-3, 6-2, 7-5 victory on Arthur Ashe Stadium to repeat his opening round triumph over Dimitrov at Wimbledon.
“The last time I played on this great court, I won the title so it was great to be able to come back and play again,” said the 33-year-old Swiss.
Wawrinka, the former world number three, now down at 101, shrugged off having to take a medical time-out early in the third set.
South African fifth seed Kevin Anderson, the 2017 US Open runner-up to Nadal, also needed a medical timeout for a leg injury in the third set of his clash with Ryan Harrison of the United States.
But Anderson, defeated in the Wimbledon final last month by Novak Djokovic, recovered to win 7-6 (7/4), 5-7, 4-6, 6-3, 6-4.
Denis Shapovalov, the 19-year-old 28th seed, was 7-5, 5-7, 4-1 ahead of 18-year-old Felix Auger-Aliassime when his Canadian teammate and close friend was forced to retire.
Auger-Aliassime, making his Slam debut after coming through qualifying, had needed a medical timeout for an irregular heartbeat in the third set.
Greek 15th seed Stefanos Tsitsipas saw off 36-year-old Tommy Robredo of Spain 6-3, 7-6 (7/1), 6-4.
Tsitsipas, 20, made the Toronto Masters final this month seeing off the likes of Novak Djokovic and Alexander Zverev on the way before losing to Rafael Nadal.
The Greek was only three years old when Robredo played his first US Open in 2001.
US hopes were buoyed by wins for 11th seed John Isner and 18th-ranked Jack Sock who won his first singles match since May.