Federer, the second seed, trounced Indian qualifier Somdev Devvarman, ranked 188 in the world, 6-2, 6-1, 6-1 and next faces France's Julien Benneteau.
It was Federer's most one-sided win in Paris since he beat Belgium's Kristof Vliegen in the first round in 2004.
The Swiss great's latest French Open victory took him to 56, just two behind the tournament record held jointly by Guillermo Vilas and Nicola Pietrangeli.
"It's great that all the work is paying off and I hope to keep experiencing this as long as possible," he said.
Federer, the 2009 champion and bidding for an 18th major, beat Devvarman in their only previous meeting in straight sets in Dubai in 2011.
Tsonga, the sixth seed, cruised past Finnish veteran Jarkko Nieminen, 7-6 (8/6), 6-4, 6-3.
Tsonga, a quarter-finalist last year when he squandered four match points against Novak Djokovic, next faces compatriot Jeremy Chardy.
"I'm French, it's in France, and of course there may be a bit more pressure, but for me it's positive," said Tsonga.
Wildcard Monfils, a semi-finalist in 2008, won a thrilling contest against in-form but unpredictable Latvian Ernests Gulbis, 6-7 (5/7), 6-4, 7-6 (7/4), 6-2 and next faces Spanish veteran Tommy Robredo.
Monfils was playing his 12th match in 17 days after reaching the final in Nice last weekend.
"I don't think about fatigue, I just hope to be ready for my next match," said the 26-year-old who hit 29 winners and 33 unforced errors.
Gulbis finished with 52 winners but an ugly 69 errors.
In a match played in a fine spirit, Monfils even whipped his mobile phone from his bag at the end of the third set to film the crowd giving him a standing ovation after he had recovered from squandering five set points in the ninth game.
Spanish men's fourth seed David Ferrer, a semi-finalist in 2012, also made the third round, beating compatriot Albert Montanes, 6-2, 6-1, 6-3.
In the women's event, double Australian Open champion Victoria Azarenka cruised into the second round.
Third-seeded Azarenka clinched a 6-1, 6-4 win over Elena Vesnina — her fifth win in five matches against the experienced Russian — and will next face German 19-year-old Annika Beck, the 2012 junior champion, for a place in the third round.
Polish fourth seed Agnieszka Radwanksa, who has never got beyond the fourth round, eased past Mallory Burdette of the United States 6-3, 6-2.
She had been scheduled to tackle her sister for a place in the last 16, but Urszula Radwanska, who put out Venus Williams in the first round, crashed out 6-3, 6-3 to German qualifier Dinah Pfizenmaier, the world number 127.
Danish 10th seed Caroline Wozniacki, a former world number one, who was a quarter-finalist in 2010, also lost 7-6 (7/2), 6-3 to Serbian world number 47 Bojana Jovanovski.
Women's top seed Serena Williams, who is on a 25-match winning streak, later faces French teenager Caroline Garcia.
Williams is looking for a second title in Paris, 13 years after her first.