Fifth seed Murray recovered from a slow start to beat his Czech opponent 4-6 6-3 6-2 as the top seeds finally took to the courts in the Californian desert after receiving first round byes.
Briton Murray won 69 per cent of his second serves, compared to just 39 per cent by Rosol, which proved a key difference.
"I got broken three times in a row the end of the first set, beginning of the second," Murray told reporters.

"I just kind of kept going and found a way to win, which is always the most important thing," he said.
Only five active players have won more tour-level hardcourt matches, headed by Roger Federer (575).
Seventh seed Federer dominated the first set but was taken to a tie-break in the second before prevailing 6-2 7-6(5) against French qualifier Paul-Henri Mathieu, while his Swiss compatriot, and Australian Open champion, Stanislas Wawrinka clinched a 6-3 7-5 win over Croatia's Ivo Karlovic.
Australian Lleyton Hewitt, however, failed to notch his 600th career win when he was beaten by South African Kevin Anderson 7-6(5) 6-4.
The veteran Hewitt saved 10 of 11 break points but was unable to break the tall South African at all, which proved the difference in the match.
There was high drama in another match as Colombian Alejandro Falla saved a match point at 5-2 down in the third set to beat Polish 18th seed Jerzy Janowicz 6-3 2-6 7-6(5).
On the women's side, top seed Li Na beat Chinese compatriot Jie Zheng 6-1 7-5. Australian Open champion Na notched six breaks as Zheng struggled on serve.


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