Norrie joined Andy Murray, Jack Draper and Dan Evans in round three, making it the first time four British men have gone this far in New York since the start of the professional tennis era in the 1960s.
It is also only the second time in the open era that four British men have reached the third round at any grand slam after Wimbledon 25 years ago.
Of the quartet, Norrie went into the tournament as the banker given his incredible consistency and top eight seeding, and he has not disappointed.
After keeping a cool head against the wavering attention span of Benoit Paire in round one, Norrie proved too strong for Portugal’s Sousa in a more conventional contest, winning 6-4 6-4 7-6 (4).
The only moment of concern came late in the third set, when Norrie was forced to save a set point at 5-6, but he kept his nose in front in the tie-break to move safely through.
The victory means the seventh seed, who next faces young Dane Holger Rune, has already matched his best performance in New York with hopefully significantly more to come.
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 Australian 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 said: “It was getting difficult. I was in control and then I thought he played very well to get back in.
“I was just happy to get off, really. It was nice to win that match to get to the third round. It’s a match I’m on paper supposed to win. It was my goal to get to where I needed to get to in the seeding spots, do my job, and then go from there.”
Evans had a lot of support, including from Tim Henman, who was watching courtside.
The pair have very similar game styles, and Evans said: “I speak to him quite often. He’s obviously a reassuring figure to have by the side of the court. He’s close probably to getting me a coaching fine pretty quickly.”
Evans will now take on former champion Marin Cilic for a place in the last 16.
Another historic moment will come after the tournament when Draper will join Norrie, Evans and Murray in making it the first time since the formalisation of the rankings in 1973 that there are four British men in the top 50.
The 20-year-old began the season outside 250 but has climbed at a phenomenal rate and gave the most exciting demonstration yet of his potential by defeating sixth seed Felix Auger-Aliassime in straight sets on Wednesday.
He is reaping the benefits of having more experienced players to learn from and look up to, and he said: “I think I’ve known my tennis ability has been good for a long time now.
“I’ve obviously had the privilege to hit with Andy, Cam, Dan. They’re top players in their own right. I know I could hold my own a while back with them.
“It’s more been about how am I going to cope mentally and physically at this level and do it consistently. That’s what top-tier tennis is all about, being able to be there every single point, compete with these guys.
“It’s mad really. My target at the start of the year was two things: stay injury-free and try to break the top 100. Now I’m going to be top 50.
“I’m obviously very proud of myself. It’s a nice thing for all my hard work that I’ve put in on a daily basis with the people that care about me and support me. Hopefully onwards and upwards from here.”