The British number two has only once failed to make the last 32 when he has been in the main draw and progressed to the fourth round 12 months ago.
Having lost only six games to Jiri Vesely in the first round, Evans looked set for another comfortable afternoon against the 30-year-old Duckworth when he eased into a two-set lead.
But Duckworth, ranked 83rd, took his first break point of the match in the ninth game of the third set and then had an early break in the fourth before Evans fought back to clinch a 6-3 6-2 4-6 6-4 victory.
The 20th seed did not hide his delight at the end, spreading his arms wide and running to the net after taking his first match point with a backhand pass threaded down the line.
Evans will now take on former champion Marin Cilic for a place in the last 16.
Meanwhile, Iga Swiatek laid down a marker with a straight-sets victory over former champion Sloane Stephens.
It looked to be a tricky draw for the world number one, who has struggled for form since her 37-match unbeaten run ended at Wimbledon.
But she has been untroubled so far at Flushing Meadows and eased to a 6-3 6-2 victory on Arthur Ashe Stadium.