Williams, aiming for a 22nd Grand Slam singles title, thrashed Magdalena Rybarikova 6-2 6-0 in just 42 minutes.
Djokovic, a defeated finalist in three of the last four years, beat Chinese Taipei's Yen-Hsun Lu 6-4 6-1 6-1.
Earlier, Australian Open champion Angelique Kerber was beaten by world number 58 Kiki Bertens.
Fifth seed Victoria Azarenka was another high-profile casualty, forced to retire with a thigh injury when trailing Italy's Karin Knapp 4-0 in the third set.
But nine-time champion Rafael Nadal advanced at speed and in style, taking only one hour and 20 minutes to overcome Sam Groth 6-1 6-1 6-1.
The 29-year-old Spanish fourth seed, who hit an astonishing winner through his legs in the third set, will face Argentine world number 99 Facundo Bagnis in the second round after taking his career French Open record to 71 wins and two losses.
Djokovic will face Belgian Steve Darcis next, ahead of a potential third-round encounter with British number two Aljaz Bedene in the third round.
Djokovic, who won bronze at the Beijing 2008 singles and came fourth at London 2012, believes ranking points should be on offer at the Olympics.
The Olympics is organised by the International Tennis Federation, which has not reached an agreement with the men's and women's tours to recognise players' performances at Rio 2016.
John Isner admitted that the lack of points on offer was "a pretty big factor" in his decision not to represent the United States at the Games. Australian Bernard Tomic and Austria's Dominic Thiem have also made themselves unavailable to their countries.
"We have the best players in the world participating in arguably the fifth Grand Slam. It's of that importance for all of us, even more, because it happens every four years," Djokovic said.
"I would definitely encourage people to rethink getting points out there."
Third seed Kerber, who has yet to advance past the quarter-finals at Roland Garros, lost 6-2 3-6 6-3.
The German, 28, lost her opening matches at the Madrid and Rome tournaments and a shoulder injury ruled her out of Nuremberg last week.
Bertens, from the Netherlands, won the title in Nuremberg.
"First rounds are always tough in the tournament, especially for me. What can I say? It happens," said Kerber, who stunned Serena Williams in Melbourne to win her maiden Grand Slam title earlier this year.
Eugenie Bouchard is now down at 47 in the rankings but her form of late has been a little more encouraging and she eased past Germany's Laura Siegemund 6-2 6-2.
The Canadian had a difficult season in 2015 after reaching the top 10 the previous year and found the whole experience so stressful it led to an eating disorder.
She said: "I felt a lot of pressure and kind of this expectation that if you win a match it's normal and if you lose it's a disaster.
"Before matches I was very nervous and definitely had trouble eating. I just felt like it would come right back up. It's a difficulty I went through. And not just before matches, but it happened to me at other meals as well.
"So people think I was concerned about my body image and things like that and I was losing weight on purpose. That was really not the case. I was just so stressed."
Francesca Schiavone had to deny she was retiring after Roland Garros' Twitter feed mistakenly announced this as her final appearance.
The 35-year-old Italian, who won the title in 2010 and reached the final again the following year, was given a prolonged ovation after her 6-2 6-4 loss to France's Kristina Mladenovic.
"I appreciate this situation," said Schiavone. "But I think that everybody thought this because Roland Garros announced it. It was not the last one for me."
Subscribe to the BBC Sport newsletter to get our pick of news, features and video sent to your inbox.
Top seeds Serena Williams and Novak Djokovic eased through to the French Open second round with comprehensive victories.