British No. 1 Norrie was barely troubled in a convincing 6-4 6-4 7-6(4) win over unseeded Portuguese Joao Sousa.
Meanwhile, Evans, who is seeded 20th at the Grand Slam, defeated Australia’s James Duckworth in four sets, 6-3 6-2 4-6 6-4.
Murray, who won his maiden major crown in New York a decade ago, reached the third round of a Grand Slam for only the second time in five years.
It is the first time that the 35-year-old, who plays Italian 13th seed Matteo Berrettini next, has reached the round of 32 at the US Open since 2016.
Seventh seed Norrie, who reached semifinals at Wimbledon in July, matched his career-best run at the US Open with his victory over Sousa.
“I think it’s very cool to have four of us through and especially all of us supporting each other. It’s great,” Norrie said.
“We’re all feeding off each other and it’s good because we’ve got a great team leading into Davis Cup and we’re all in form, all playing well and enjoying our tennis.”
“I don’t see why we can’t keep going.”
The victories mark only the second time in the Open era that four British men have progressed to the third round at a Grand Slam, and the first time at the US Open, following Tim Henman, Greg Rusedski, Mark Petchey and Andrew Richardson’s progress to the Wimbledon third round in 1997.
Information from Reuters contributed to the report.