Third seed Wawrinka fired 56 winners but committed 46 unforced errors and saved eight break points to book a second round clash against Japan's Taro Daniel.
Wawrinka, 31, had defeated 59th-ranked Rosol in the semi-finals of the Geneva clay-court tournament just last Friday.
But Rosol went into the match on a chilly Court Philippe Chatrier with the pedigree of having famously stunned Rafael Nadal at Wimbledon in 2012.
That free-swinging masterclass almost paid dividends again Monday but once he had let slip two break points at 2-2 in the fourth set, his hopes slowly slipped away.
Muguruza, who has made the quarter-finals for the last two years, will take on French wildcard Myrtille Georges, the world number 203, for a place in the last 32.
But the 22-year-old Wimbledon runner-up was far from impressive in the two and a half hour encounter.
After dropping the first set, Muguruza had to save nine break points just in the opening game of the second set.
She then allowed her 37th-ranked opponent, who has won just one match all year, to claw her way back from 0-4 down to 3-4 in the decider before the Spaniard settled herself to see out the win.
However, there was plenty of time for notable firsts. Qualifier Cagla Buyukakcay beat Aliaksandra Sasnovich of Belarus 5-7, 7-6 (7/2), 6-2 to become the first Turkish woman in the Open era to reach the second round of a Grand Slam.
The 26-year-old Buyukakcay, ranked at 83, had earlier this clay-court season become the first Turk to win a WTA title when she triumphed in Istanbul.
Elsewhere, Canadian eighth seed Milos Raonic, a 2014 quarter-finalist, defeated Serbia's Janko Tipsarevic 6-3, 6-2, 7-6 (7/5). Tipsarevic, ranked at 686, was playing his first Roland Garros since 2013 after being treated for a benign tumour on his left foot.
Japanese fifth seed Kei Nishikori beat Italy's Simone Bolelli 6-1, 7-5, 6-3 on Court Philippe Chatrier on Sunday.
Victory gave the 26-year-old Japanese his 50th Grand Slam win. Russia's Svetlana Kuznetsova, the 2009 champion, won her resumed match against Yaroslava Shvedova of Kazakhstan 4-6, 6-1, 6-4.
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