Serbian world number one Djokovic wasted no time on Court Suzanne-Lenglen, defeating his Swiss rival 6-1, 6-4, 6-3 in just 93 minutes.

Djokovic, who will hold all four Grand Slam titles for the second time in his career if he wins the tournament, will next face Italy's Salvatore Caruso, who defeated Frenchman Gilles Simon 6-1, 6-2, 6-4. 

Top seed in the women's draw Naomi Osaka had a much sterner test as she battles to hold on to her place at the top of the world rankings.

In a gruelling match with Victoria Azarenka, the Japanese star found herself a set and a break down as the Belarussian showed glimspes of the talent that earned her world number one status in 2012.

Osaka, targeting a third consecutive Grand Slam title having won the US Open and Australian Open, trailed 4-6, 3-4, with Azarenka breaking serve to close on a straight sets victory.

But Osaka dug deep to get back on serve and took the set 7-5 to level the match before normal service resumed and she closed out the match with a 4-6, 7-5, 6-3 victory.

Women's third seed, Romania's Simona Halep, came through a rigorous battle with Poland's Magda Linette, eventually cruising through the third set to record a 6-4, 5-7, 6-3 win.

The biggest upsets of the day came in the women's draw with 11th seed Aryna Sabalenka, 16th seed Qiang Wang, 21st seed Daria Kasatkina and 24th seed and home favourite Caroline Garcia all losing.

Belarussian Sabalenka was the first of the seeds to tumble as she lost in straight sets against American Amanda Anisimova 6-4, 6-2.

Wang of China struggled with her first serve (54%) and Poland's Iga Świątek, ranked 104th in the world, was able to capitalise as she stormed to a 6-3, 6-0 victory.

Despite racing into a 6-1 first set lead, French favourite Garcia was usurped in a thriller by Russia's Anna Blinkova.

Scoring on 82% of her first serves, the 117th-ranked player levelled the match at 1-6, 6-4, before edging the deciding set 6-4 to set up a third round clash with Monica Puig of Puerto Rico.

Tenth seed Serena Williams eased into round three with a 6-3 6-2 victory against Japanese qualifier Kurumi Nara and eighth seed Ashleigh Barty from Australia had a comfortable 7-5 6-1 triumph against American Danielle Collins.

Having smashed his racquet in his first round victory against Australian John Millman, things were much calmer for Germany's Alex Zverev as he eased to a 6-1, 6-3, 7-6 triumph against Mikael Ymer of Sweden.

Fourth seed Dominic Thiem, from Austria, needed four sets to see off Kazakhstan's Alexander Bulbik 6-3, 6-7, 6-3, 7-5. 

Argentine Juan Martin Del Potro needed five sets to overcome a spirited effort from Yoshihito Nishioka.

The eighth seed was made to work for his place in round three and eventually saw off his Japanese rival 5-7, 6-4, 6-2, 6-7, 6-2 in a contest that lasted 3 hour 46min.