The Scot, 29, beat Rosol of the Czech Republic 6-3 6-2 6-2 and will play Spain's Marcel Granollers in round two.
Murray joined compatriots Dan Evans and Naomi Broady as a winner on day two, with Kyle Edmund and Johanna Konta already into the second round.
Heather Watson and Aljaz Bedene both suffered first-round losses on Tuesday.
Watson struggled with illness as she lost 6-2 7-5 to Dutch qualifier Richel Hogenkamp, and has now failed to reach the second round at Flushing Meadows in six attempts.
Bedene was beaten 6-4 6-4 6-4 by Australian Nick Kyrgios.
Broady had earlier beaten compatriot Laura Robson 6-7 (4-7) 6-3 6-4 to claim only the second Grand Slam main-draw win of her career.
Evans began Tuesday's run of British success as he saw off American Rajeev Ram 6-2 4-6 7-5 6-1.
