Second seed Murray dropped the first two sets 6-3, 6-3 before taking the third 6-0.
He then edged ahead with a break in the fourth at 4-2 when play was halted in the gloom of Court Philippe Chatrier at 2123 (1923GMT).
At 37, Stepanek is bidding to be the oldest winner of a singles match in Paris since Jimmy Connors in 1991.
And he appeared to be cruising to a shock win over the three-time semi-finalist when he swept the first two sets.
But Murray’s greater stamina and his eight-year age advantage resulted in a dramatic turnaround as he took the third in just 18 minutes.
In between, the Scot was warned for swearing while Stepanek was also hit with a code violation for time wasting as he desperately demanded that the tie be called off for the night after the third set.
But he was ordered to keep playing and when the tie was halted it was Murray in the ascendancy ahead of a Tuesday resumption where he will attempt to avoid his first opening round exit in Paris since his 2006 debut.
Wawrinka needed five tough sets to get past Stepanek’s equally combative and fiesty Czech compatriot, Lukas Rosol.
Third seed Wawrinka fired 56 winners but committed 46 unforced errors and needed to save 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 faded.
“He’s typically very aggressive and puts a lot of pressure. That’s what he did today,” said Wawrinka of a match played out in temperatures of just 12 degrees.
“I was very happy to pick up the fifth set in spite of the cold.”
– Italian veterans out –
Japan’s fifth seed Kei Nishikori reached the second round with a 6-1, 7-5, 6-3 win over Simone Bolelli of Italy.
Nishikori goes on to face Russia’s Andrey Kuznetsov after finishing off a first-round tie which had been suspended overnight with him leading 2-1 in the third set.
It was a 50th career Grand Slam match win for Nishikori who made the quarter-finals in Paris in 2015.
Tenth-seeded former US Open champion Marin Cilic, however, suffered his earliest loss at a Slam in five years when Argentine qualifier Marco Trungelliti clinched a shock 7-6 (7/4), 3-6, 6-4, 6-2 win.
It was only the fourth career victory for the 26-year-old world number 166.
Spanish fourth seed Garbine Muguruza struggled on Court Suzanne Lenglen, coming back from a set down to defeat Slovakia’s Anna Karolina Schmiedlova 3-6, 6-3, 6-3.
Muguruza, who has made the quarter-finals for the last two years, will next take on French wildcard Myrtille Georges, the world number 203.
Muguruza fired 44 winners but hit 53 unforced errors and had to save 17 of 21 break points.
Seeds falling by the wayside included seventh seed Roberta Vinci, the Italian who stunned Serena Williams at the US Open semi-finals last year.
The 33-year-old suffered a third successive first-round exit in Paris going down 6-1, 6-3 to Kateryna Bondarenko who hadn’t won a match at the French Open since 2010.
Vinci’s Italian 16th seeded compatriot Sara Errani, the 2012 runner-up, lost to 2010 Wimbledon semi-finalist Tsvetana Pironkova 6-3, 6-2.
Second seed Agnieszka Radwanska of Poland, however, brushed aside Serbia’s Bojana Jovanovski 6-0, 6-2.
Jovanovski, the world number 120, is still without a win in 2016.
Romanian sixth seed Simona Halep, the 2014 runner-up, took just 43 minutes to defeat Japan’s Nao Hibino 6-2, 6-0.
Play started two and a half hours late because of rain with 12 of the 66 scheduled matches pushed back to Tuesday.
Another six were later suspended.
However, there was time for a notable first.
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.
Canadian eighth seed Milos Raonic, a 2014 quarter-finalist, defeated Serbia’s Janko Tipsarevic 6-3, 6-2, 7-6 (7/5).
Tipsarevic was playing his first Roland Garros since 2013 after being treated for a benign tumour on his left foot.