The 31-year-old Serb, seeded 20th, went behind at the start of the first two sets but recovered well to overcome world No134 Dutra Silva.
Djokovic, who has not won a grand slam since his victory at Roland Garros in 2016, will face Spain's Jaume Munar in the second round.
Another former champion Stan Wawrinka, who has struggled with a persistent knee injury, was beaten 6-2 3-6 4-6 7-6(5) 6-3 by big-hitting Spaniard Guillermo Garcia-Lopez.
World No1 Rafa Nadal, chasing a record-extending 11th French Open title, took a two-set lead but was a break down in the third against Italian Simone Bolelli before rain halted play.
The pair will return on Tuesday with Nadal leading 6-4 6-3 0-3.
In the women's draw, world number two Caroline Wozniacki overcame a slow start to beat American Danielle Collins 7-6(2) 6-1.
Twice Wimbledon champion Petra Kvitova came back from a set down to beat Paraguayan Veronica Cepede Royg 3-6 6-1 7-5.