Second seed Murray was the last of the five to progress with a late-night 6-3 6-2 6-2 victory over Rosol
Of the eight British players in the main draw at the US Open, the most since 1985, five made it through to round two, with Andy Murray, Dan Evans and Naomi Broady joining Johanna Konta and Kyle Edmund in the last 64 on Tuesday.
The quintet of survivors matches the best performance by British players at Flushing Meadows since 1987, when four women and one man made it through the first round.
Second seed Murray, looking ominous in all black, was the last of the five to progress with a late-night 6-3 6-2 6-2 victory over Czech Lukas Rosol, effortlessly advancing without facing a single break point.
"I served very well," said Murray, who set up a second round encounter against Marcel Granollers of Spain.
"It was a tough start to the match, he had a few chances early on but once I got the break-up, I played well."
Evans was ranked as low as 772 in May 2015 and still outside the top 300 when the U.S. Open began last August.
A string of successes on the Challenger Tour carried him close to the top 100 and his 6-2 4-6 7-5 6-1 over the American Rajeev Ram is set to lift his ranking from 64 to inside the top 60, a new career-high.
"I would have taken your hand off," Evans responded when asked how he would have reacted had someone offered him a chance to be ranked this high at this stage of the year.
"It's been a pretty interesting year and I'm happy with how it's been."
Evans next plays 27th-seeded German Sascha Zverev, who at 19 is tipped as a future star of the men's game.
World number 82 Broady earned only her second ever grand slam win with a gutsy 6-7 6-3 6-4 win over fellow Briton Laura Robson.
The 26-year-old Broady recently broke into the world's top 100 and admitted she was a late bloomer.
"I think in British tennis they like to say that you're running out of time and you're playing catch-up and if you're not there by 18 then you're behind everybody else," she said.
"But tennis is becoming more of a longevity sport there and the average age of the top 100 is getting older.
"I think I was 25 when I broke into it this year. I'm still there and I maybe won't be in the top 100 as long as Laura has the potential to be but we've not got the same career, so I'll just do the best with what I've got."
There was disappointment for Heather Watson, however, who suffered from a fever and back pain as she lost 6-2 7-5 to Dutch qualifier Richel Hogenkamp.
The other British player in action, Aljaz Bedene, also suffered a straight sets defeat, with Australian 14th seed Nick Kyrgios proving too strong in a 6-4 6-4 6-4 victory.