Murray is arguably favourite in New York to claim his fourth grand slam title and the Scot justified the tag, thrashing Rosol 6-3 6-2 6-2.
He will now face world number 45 Marcel Granollers after the Spaniard had earlier beaten Argentina’s Juan Monaco in straight sets.
Murray’s progress means Britain will have five representatives through to round two at Flushing Meadows for the first time in 29 years, with Kyle Edmund, Dan Evans, Johanna Konta and Naomi Broady also winning their opening encounters.
Edmund’s shock victory over world number 15 Richard Gasquet was the pick of the bunch, and Murray was impressed with the 21-year-old’s display.
“I saw the first couple of sets of Kyle’s match, that’s been it,” Murray said.
“I didn’t get to see any of the matches (on Tuesday). Kyle played extremely well. I practised with him the day beforehand. He was hitting the ball good in practice.
“He’s improving all of the time. To win a match like that in a slam that comfortably against a top player, a guy that’s been at the top for a long time is a very good sign. It was good for him.
“Then obviously all the other Brits, obviously Naomi and Laura (Robson) had a tight match. Dan got through a tricky one against Rajeev Ram.
“Konta has been solid for a long time. Heather (Watson) has never played so well here. It’s been I guess a pretty good start for the Brits.”
Novak Djokovic’s injury-hampered win over Jerzy Janowicz on Monday has only fanned the hype around Murray’s challenge, but the level of expectation does not seem to be weighing heavy on the world number two.
With coach Ivan Lendl back in his box for the first time since Wimbledon, Murray did not have to defend a single break point against Rosol, while capitalising on five himself.
“At the start of the match he had a few chances the first couple of service games. He came out going for his shots,” Murray said.
“I got through that tricky period right at the start where he’s hitting the ball really well, kind of adjusting to the conditions.
“Once I got through that, I settled down and played I think a really good match.”
Rosol is best known for his shock Wimbledon defeat of Rafael Nadal in 2012, but that has been his only success against top-five ranked players now in 15 attempts.
The straight-talking Czech did raise eyebrows in Munich last year when he bumped into Murray at a change of ends, prompting the Scot to tell his opponent: “No-one likes you on the tour, everyone hates you.”
Those hoping for a re-run, however, would have left disappointed as Murray coasted through a rather sleepy contest in Arthur Ashe Stadium, during which the favourite was in complete control throughout.
Granollers, ranked 45th in the world, is likely to offer a sterner test.
“He plays different to a lot of players now, he likes to come forward a lot, he’s got great hands and has good feel up at the net,” Murray said.
“He doesn’t give you much rhythm so hopefully I’ll play a good match.”