The 25-year-old had never got past round two of a Grand Slam tournament before entering the US Open.
But the Johannesberg-born player, who pulled off a shock by knocking out 13th seed Diego Schwartzman in the first round, battled past Argentinian Coria 6-3, 6-4, 6-4 in a close two hours and seven minutes contest.
Unlike in his first match where he had to fight back from the brink, Norrie took control of the first two sets against a jaded-looking Coria.
The 28-year-old – younger brother of former star Guillermo – showed some resistance in the third set but missed numerous opportunities to convert break points at 4-3 up and Norrie punished him for his failure to capitalise.
Norrie is the first Briton to make into round three after Kyle Edmund lost in four sets to Novak Djokovic.
He will face Alejandro Davidovich Fokina, the unseeded Spaniard who knocked out 24th seed Hubert Hurkacz in the second round.
Andy Murray faces a tough challenge on Friday against rising Canadian star Felix Auger-Aliassime, 19, while ninth seed Johanna Konta is up against Sorana Cirstea of Romania.
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