Cilic, a former US Open champion who reached two of the past three Grand Slam finals, was not bothered by the rain that stopped play for about an hour and won 6-3, 7-5, 7-6 (4).
Shapovalov compiled a 32-9 edge in winners and converted 6 of 12 break points.
The Swede lost his ground and Nadal broke him at the start of the final set to control the scoreboard in the rest of the match and reach the second round when Robin netted a forehand in the last point of game seven to earn one of his most impressive records on clay.
Mixing deep groundstrokes with subtle shots, Bolelli obtained another set point with a backhand drop shot to lead 7-6, but Nadal hit a service victor on the next point.
After getting rained off the schedule on Monday night, two-time champion Maria Sharapova made a comeback to the tournament after missing the 2016 and 2017 editions.
Nadal won the first three games on the day but the 129th-ranked Bolelli raised his level and managed a series of good returns. "He had many chances in the third set", Nadal said.
Everton defender Ashley Williams taken to hospital during Wales' draw in US
Wales, who missed out on qualification for the World Cup, were playing without Gareth Bale after he helped Real Madrid to the Champions League trophy on Saturday and managed just one shot on goal .
Michael Bisping announces retirement from UFC due to sight issues
Bisping debuted in the UFC as the victor of The Ultimate Fighter season 3 at light heavyweight. Now it's time for Michael to achieve greatness in the next phase of his professional career.
Pippa Middleton Is Radiant In Red At The 2018 French Open
She wore a stunning red wrap-style maxi dress by Ralph Lauren which gave away only the slightest hint of her bump. It's not just Kate Middleton's style we're the envy of, because her sister is also quite the style icon.
"It was a very hard match, Simone played very aggressively, he had many chances in the third set", Nadal said.
The 31-year-old, five-time major champion, took just 24 minutes to win the first set of a match which was delayed from Monday due to thunderstorms, but threw away a 3-1 lead in the second.
After beating Serena Williams in the 2016 final, Muguruza's title defence was ended in the fourth round a year ago.
Yuki Bhambri's maiden appearance at the French Open ended with a first round defeat even as Rohan Bopanna began his men's doubles campaign with an easy win along with French partner Edouard Roger-Vasselin.
The American insisted it was "fun as well as functional" in a nod to the severe health problems she suffered after giving birth to daughter Alexis Olympia in September.