Nadal had left his participation in the singles in doubt until the last minute because of a thigh injury, and was made to work hard for his 6-1, 4-6, 6-4 victory.
The 38-year-old looked to be cruising after racing through the first set on Court Philippe Chatrier, the site of his 14 French Open titles, before Fucsovics dug his heels in.
The Hungarian world number 83 raised his level after a poor start and forced Nadal to a third set.
Fucsovics held three break points for a 3-1 lead in the decider but Nadal withstood the threat, breaking in the following game to wrest the momentum back.
A gritty hold of serve edged Nadal closer to victory, with the 22-time Grand Slam champion eventually seeing off a dogged Fucsovics after two hours and 30 minutes.
Nadal, a singles gold medallist at the 2008 Beijing Olympics and doubles champion at Rio four years later, is playing just his seventh tournament of 2024.
He reached the final in Bastad last weekend but suffered a setback in training on Wednesday, jeopardising his hopes of appearing at a fifth Olympics.
Nadal teamed up with Carlos Alcaraz in the doubles on Saturday, but he had his right thigh heavily bandaged as the pair won their first-round match in straight sets.
