Broady, seeded 13 in the singles competition, saw off Finland's Herkko Pollanen 6-1 7-5 in his early match at Flushing Meadows.
The 18-year-old from Stockport then despatched Maxime Hamou of France 6-7(5) 6-4 6-3 to book a place in the fourth round.
Compatriot Kyle Edmund also won his second round encounter, 7-6(5) 6-4 against Shotaro Goto of Japan, but the South Africa-born 17-year-old could not back that up with another victory, crashing out of the tournament 6-3 6-3 to second seed Filip Peliwo of Canada.
And there was disappointment for Joshua Ward Hibbert, the 12th seed, who lost in the second round to Yoshihito Nishioka, the Japanese winning 2-6 6-1 6-2.
