The Spaniard - top seed, world number one, reigning champion and red-hot favourite for another Roland Garros crown - had to return on Tuesday morning to finish his first-round clash against the lucky loser from Italy.
Nadal had taken the first two sets in typically ominous style but then trailed 3-0 in the third when the rain came.
Nadal broke straight back on the resumption but Bolelli did not let him have things his own way, threatening another break at 4-3.
Bolelli then forced four set points in a marathon tie-break but the 16-time grand slam winner got over the line 6-4 6-3 7-6 (11/9).
British number one Kyle Edmund stormed into the second round with a straight-sets win over Australian youngster Alex De Minaur.
Edmund, looking to build on his stunning run to the Australian Open semi-final in January, simply overpowered his 19-year-old opponent.
He broke serve in the first game of the match, took the opening set in just half an hour and went on to wrap up a fine 6-2 6-4 6-3 victory.
The early break set the tone, while another in the fifth game of the second set, after De Minaur pushed a forehand wide, prompted the youngster to take a ball from his pocket and smash it out of Court Three in frustration.
De Minaur did break back to level at 4-4, however Edmund hit back immediately and held serve to take a 2-0 lead.
It was a ruthless display from Edmund, with breaks in the first and last games of the third set capping a perfect start to his Roland Garros campaign.
Third seed Marin Cilic made short work of his first-round match, going through in straight sets against Australian James Duckworth.
The Croatian, runner-up at the Australian Open in January, shrugged off a 45-minute rain delay to run out 6-3 7-5 7-6 (7/4) winner.
South African Kevin Anderson, seeded six, was another comfortable winner as he swept past Italian Paolo Lorenzi 6-1 6-2 6-4.
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