Elsewhere, four-time champion Iga Swiatek and Elina Svitolina both eased into the last 32 with straight-sets wins.
Djokovic had to come from a set down in his opening match against Giovanni Mpetshi Perricard and was pushed hard again by another Frenchman in Royer, before finally sealing a 6-3, 6-2, 6-7 (7/9), 6-3 win after three and three-quarter hours on Court Philippe Chatrier.
“I hope I won’t play any more French players for the rest of the tournament,” said the third seed, who has been tied on 24 Grand Slam titles with Margaret Court since winning the 2023 US Open, with a wry smile.
Djokovic, who beat world number one Jannik Sinner in the Australian Open last four earlier this year, cannot meet the red-hot title favourite until the final at Roland Garros.
Djokovic powered through the first two sets, although world number 74 Royer, born just 5km from Roland Garros, showed much more resistance in the third before extending the match in a brilliant tie-break.
The Serb, who twice failed to consolidate breaks of serve in the third set, made no such mistake in the fourth as he belatedly booked his place in the next round on his fifth match point.
Czech Jakub Mensik, the only player other than Djokovic to beat Sinner this year, needed eight match points before eventually downing Mariano Navone 6-3, 2-6, 6-4, 1-6, 7-6 (13/11).
Rising Spanish star Rafael Jodar reached the third round of a Grand Slam for the first time, seeing off James Duckworth 6-1, 6-7 (5/7), 6-4, 7-5.
Meanwhile, Australian Open champion Rybakina blew a one-set lead to exit in dramatic fashion, slumping to a 3-6, 6-1, 7-6 (10/4) loss to Ukraine’s Yuliia Starodubtseva.
It is Rybakina’s earliest departure from any tour-level tournament since the 2025 Miami Open and first defeat in the opening two rounds of a major since the 2024 Australian Open.
“I couldn’t find the balance. I couldn’t find the right shot, and it was clearly just too many mistakes,” said the Kazakh world number two.
Starodubtseva will face China’s Wang Xiyu for a place in the last 16. (AFP)