The 31-year-old Serb, seeded 20th, went behind at the start of the first two sets but recovered well to overcome world number 134 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 number one 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.
Highlights from day two of the French Open tennis championships on Monday (all times GMT):
1745 PLAY SUSPENDED DUE TO RAIN
Play was suspended on all courts due to rain, with defending champion Rafa Nadal leading 6-4 6-3 0-3 against Italian Simone Bolelli on Court Philippe Chatrier.
1545 WOZNIACKI RACES INTO ROUND TWO
World number two Caroline Wozniacki moved past American Danielle Collins 7-6(2) 6-1 in the opening round.
The Dane will next face Spanish qualifier Georgina Garcia Perez, who knocked out Dalila Jakupovic 6-3 6-4 to claim her first main-draw victory at a Grand Slam.
1410 BAGHDATIS BREAKS DOWN IN TEARS
Cypriot Marcos Baghdatis suffered an emotional meltdown after he retired hurt against Santiago Giraldo of Colombia in the first round.
Baghdatis, the 2006 Australian Open finalist, was leading 6-3 4-3 when he struggled to move on court before asking for a medical timeout. The 32-year-old broke down in tears and went on to smash his racket while waiting for the physiotherapist to arrive.
After losing the first two points on his serve in the eighth game, Baghdatis decided to pull the plug on his campaign.
1400 THIEM CONTINUES FINE FORM
Seventh seed Dominic Thiem, who lifted his second title of the season in Lyon last week, raced to a 6-2 6-4 6-1 victory over Belarusian Ilya Ivashka.
The two-time French Open semi-finalist will next face promising Greek teenager Stefanos Tsitsipas.
"He's one very good, upcoming player. Going to be top 10, for sure, in the future and playing already unbelievable," Thiem said. "But, yeah, hope that I can have an advantage over best of five."
1345 DJOKOVIC ADVANCES
Former champion Novak Djokovic began his campaign with 37 winners in a 6-3 6-4 6-4 win over Brazil's Rogerio Dutra Silva.
The Serb, who lifted the Musketeers' Cup in 2016, has reached the second round for the 14th consecutive year in Paris.
1340 TRUNGELLITI SECURES MASSIVE PAYDAY
The 1,000km roadtrip from Barcelona to Paris was well worth it for Marco Trungelliti, who overcame Australia's Bernard Tomic 6-4 5-7 6-4 6-4 in the opening round and guaranteed himself a massive payday.
The International Tennis Federation are looking for a clarification from the Grand Slam Board on the rules for prize money distribution, but said the Argentine lucky loser will at least get 79,000 euros ($91,758) for reaching the second round.
1255 WAWRINKA BOWS OUT
Last year's runner-up Stan Wawrinka was beaten by big-hitting Spaniard Guillermo Garcia-Lopez 6-2 3-6 4-6 7-6(5) 6-3 in the opening round match which lasted three and a half hours.
Wawrinka, who won the French Open title in 2015, suffered a second loss to Garcia-Lopez in Paris after losing at the same stage four years ago.
1120 KVITOVA SURVIVES SCARE TO REACH SECOND ROUND
Two-time Wimbledon champion Petra Kvitova rallied from a set down to beat Paraguayan Veronica Cepede Royg 3-6 6-1 7-5 in a two-hour thriller on Court Philippe Chatrier.
When reminded of her current winning streak on clay, Kvitova said: "It's 12 matches... but I'm still so far from Rafa (Nadal) so I'll just keep trying."
1110 AZARENKA SUFFERS EARLY EXIT
Former world number one Victoria Azarenka struggled for rhythm in a 7-5 7-5 defeat by Czech Katerina Siniakova in the first round.
The 28-year-old Belarusian, who reached the semi-finals in 2013, committed 38 unforced errors in her first match at Roland Garros in two years.
1045 OSAKA POWERS PAST KENIN
Japan's Naomi Osaka fired 10 aces on her way to a 6-2 7-5 first-round victory over American Sofia Kenin. The 21st seed will next face Zarina Diyas of Kazakhstan.
(Compiled by Hardik Vyas in Bengaluru,; Editing by Ed Osmond)
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