The 34-year-old, who is looking to become the first man in more than 50 years to win all four Slams multiple times, won 6-3, 6-2, 6-4 on Court Suzanne Lenglen.
"I stayed concentrated. I thought the third set was very difficult for me because he lifted the level of his game," said the top seed after his 350th Grand Slam match.
"It's a cliche to say 'day by day', but it's very important for me to stay in the moment."
Djokovic will face Lithuanian Ricardas Berankis for a place in the second week.
The 2016 Roland Garros champion has won all three of his meetings with Berankis, including in last year's second round in Paris.
Cuevas tested Djokovic at times on Thursday with his creative shot-making, but the Serbian star saved eight of nine break points and hit 31 winners in a solid display.
Djokovic is seeded to meet 13-time winner Rafael Nadal, who he lost to in the final last October, in the last four.