Djokovic, a three-time Roland Garros finalist, faced little resistance from Taiwan's world number 95 Lu Yen-Hsun as he eased to a 6-4, 6-1, 6-1 victory.
The Serb, bidding to become just the eighth man to secure a career Grand Slam, needed just 90 minutes on Court Phillipe Chatrier to book a second-round encounter with Belgian qualifier Steve Darcis.
However, Murray was forced to recover from a two-set deficit to overcome 37-year-old Radek Stepanek 3-6, 3-6, 6-0, 6-3, 7-5 in a match lasting three hours 41 minutes in total.
The Scot duly returned to complete the job after making the crucial breakthrough at 5-all in the decider to set up a second-round encounter with 164th-ranked French wildcard Mathias Bourgue.
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"He's always been extremely difficult to play," said Murray after a ninth career comeback from two sets down.
"He was playing drop shots, hitting the ball very flat, it was very difficult to get into a rhythm. That's credit to him and the way he played."
The fourth-seeded Spaniard made quick work of 100th-ranked Sam Groth, taming the Australian's powerful serve with eight breaks in a resounding 6-1, 6-1, 6-1 win.
"It's obvious that was a good start for me," said Nadal.
"I managed to break him very quickly. It was important because against such an opponent it's not easy to take his serve."
However, Kerber became the tournament's first major casualty as Dutchwoman Bertens consigned the German third seed to a 6-2, 3-6, 6-3 defeat.
"She's a tough opponent. I was ready. But, yeah, at the end she made the important points."
Czech seventh seed Tomas Berdych, a former French Open semi-finalist, cruised to a straight-set win over Canada's Vasek Pospisil.