Second seed Federer reached the quarterfinals for the 11th time with a 6-3, 4-6, 6-4, 6-1 win over Monfils in match that resumed from Sunday.
Federer, the 2009 champion, will now face Swiss Davis Cup teammate Wawrinka for a place in the semifinals.
He raced through the final two sets on Monday in just over an hour after the last-16 tie was delicately poised at a set apiece when it was suspended in the chilly gloom of Sunday evening.
“I feel fresh and pumped up to be back in the quarterfinals again after missing out last year,” said Federer, who was beaten by Ernests Gulbis in the last 16 in 2014.
Monfils, a semifinalist in 2008, was nowhere near his swashbuckling best on Monday slipping to a break down in the first game of both sets.
At 2-1 down in the fourth, he needed the trainer to be called although the Frenchman's problems appeared more related to the cold from which he was suffering rather than any physical problem.
He lost the last five games of the contest and although he saved a match point with a trademark cute drop volley, an equally characteristic wild
forehand sealed his fate.
It was 33-year-old Federer's ninth win in 13 meetings against Monfils who had triumphed over the Swiss star on clay in Monte Carlo last month and in November's Davis Cup final.
Federer will take a 16-2 career advantage over Wawrinka into Tuesday's quarterfinal, including on clay in Rome earlier this month.
“It's special to play Stan, there aren't many Swiss players in the draw,” said Federer.
“I'm quite positive about it.”
Spanish seventh seed David Ferrer recorded his 40th career win at Roland Garros with a 6-2, 6-2, 6-4 victory over Croatian ninth seed Marin Cilic to also reach the quarterfinals.
Ferrer, the runner-up in 2013, will take on either third-seeded Andy Murray or unseeded Frenchman Jeremy Chardy for a place in the semifinals.
World number one Novak Djokovic, bidding for a first French Open to complete the career Grand Slam, faces French 20th seed Richard Gasquet.
The top seed has a 10-1 winning record over Gasquet whose only win over the Serb came eight years ago.
Djokovic will look to extend his run of successive Grand Slam quarterfinals to 24 while a win for Gasquet would make him just the third Frenchman to record 400 Tour-level match wins.
He is also looking for a place in the quarter-finals at Roland Garros for
the first time at the 12th attempt.
Murray, the number three seed and twice a semifinalist, was defending a 6-1 career record over Chardy, including a successful meeting on clay in Rome on the eve of Roland Garros.
Nine-time champion Rafael Nadal can clinch his 70th win at the French Open and reach a 10th quarter-final when he meets Jack Sock of the United States for the first time.
Victory for 22-year-old Sock would make him the first American man to reach the Roland Garros quarterfinals since Andre Agassi in 2003.
— Stanislas Wawrinka (@stanwawrinka) June 1, 2015