Nadal outclassed Dutchman Robin Haase 6-1, 6-4, 6-3 in one hour and 49 minutes on Court Philippe Chatrier to continue his quest for an incredible 10th Roland Garros triumph.
"I was more or less under control during the whole match, so I'm very happy. Winning in straight sets is always good news," said Nadal.
"I think I did a lot of things well this afternoon."
The Spaniard will face Nikoloz Basilashvili for a place in round four after the Georgian matched his best ever run at a Grand Slam.
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The second seed plays Argentina's Diego Schwartzman for a place in the last 16.
"I think the first two sets went very well, the third was was more difficult," said Djokovic.
"Generally, I felt better than the first round, so that's exactly what I wanted."
Sixth seed Dominic Thiem charged into round three with a routine win over Simone Bolelli, sweeping the Italian aside 7 -5, 6-1, 6-3.
The Austrian, who is the only player to defeat Nadal on clay this season, will meet American 25th seed Steve Johnson next as he looks to build on last year's run to the semi- finals.
Serving to stay in the contest when darkness halted their tie on Tuesday, Tsonga was broken immediately when play resumed as the Argentine world number 91 wrapped up a 7-5, 6- 4, 6-7 (6/8), 6-4 win.
Reigning champion Muguruza survived another tough examination from 53rd-ranked Anett Kontaveit, who beat the Spaniard in Stuttgart last month.
Muguruza has endured a difficult 2017 and again found herself in trouble after falling a set and a break behind against the Estonian.
"I knew it was going to be a battle out there," said Muguruza, who beat Serena Williams in last year's final.
"I knew that losing to her not long ago I did something different this time. I think something different in a good way.
"I have a different spirit here, I'm going to fight for every ball to try and win," she added.
- Wozniacki's double bagel -
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American qualifier Bethanie Mattek-Sands halted Petra Kvitova's comeback, defeating the popular Czech 7-6 (7/5), 7-6 (7/5) in her first tournament since a horrifying knife attack in December.
"But I really fought. I mean, after the match, I didn't feel that bad as normally I probably do.
"I'm still happy I'm here and playing. The fairytale ended. Now, in upcoming weeks, I think it will be business as usual."
Venus Williams shook off a slow start to ease into the third round with a 6-3, 6-1 victory over Japan's Kurumi Nara.
Williams, the 2002 runner-up who turns 37 next month, will play Belgium's Elise Mertens in the third round.
"Do I sense (the draw) is open? I just am really focused on me and nothing's a given, or else I'd win every match," said Venus, who has sister Serena in Paris supporting her.
The 20-year-old blamed the exertions of coming through three qualifying rounds in order to reach the main draw.
"I wasn't impressed at all," Abanda retorted when asked if she was struck by anything in Wozniacki's game.
"As I told you, fitness-wise and physically, I was tired.
"I think that she played her second match. I played five matches.