The 2015 champion struggled to time the ball on an overcast Court Suzanne Lenglen, frequently spraying his trademark backhand drive long or into the net as he looked to keep the points short against 152nd-ranked Kovalik, who was making his Roland Garros debut.
But the Swiss did enough to win the rallies that mattered, extricating himself from a deep hole in the second set tiebreak, taking the last five points after trailing 6-3.
After connecting well with a backhand to take the final point, a sigh of relief as he approached the net suggested he was glad to put the encounter - lasting a shade under two hours and in which he made 33 unforced errors - behind him.
Having had less time than he might have wished to adjust between Geneva, where he took the title on Saturday, and Paris, he took the longer view in upbeat comments after the match.
"It wasn't particularly easy after Geneva... but I feel good," Wawrinka said courtside. "I hope to go deep (into the tournament) this year."
The Swiss will play Ukraine's Alexandr Dolgopolov in the second round.
(reporting by John Stonestreet; editing by Richard Lough)
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