Having leveled their match at one set apiece before play was suspended on Sunday, Monfils lasted just over an hour after play resumed Monday before losing 6-3, 4-6, 6-4, 6-1 to the 17-time Grand Slam champion.
It's the fourth time Federer has beaten Monfils at Roland Garros, and means the Swiss star is back in the French Open quarterfinals after losing at this stage last year.
With clear skies replacing the dark clouds and rain that hit Sunday's schedule, the 2009 champion broke immediately, jumped to a 4-1 lead in the third set and never looked back to progress to the quarterfinals for the 11th time at the clay-court Grand Slam.
"I'm pumped up to be in the quarters again," said Federer, who lost in the fourth round to Ernests Gulbis in 2014. "Last year, I missed it by a hair losing in the fifth, so it's nice to be back in the quarters now."
Federer had lost his previous two matches on clay against Monfils and his encounter with the French showman was his biggest test so far in this year's tournament. He passed it in style, although he was not happy with his play at the net in the first two sets.
"Today I was just really focused on my concentration to really not miss (my volleys), and everything worked out very well," he said. "I was able to mix up my game, serve big when I had to. It was exactly what I wanted to do today."
Federer next plays his Davis Cup Swiss teammate, Stan Wawrinka, whom he leads 16-2 in head-to-head matches.
"So there will be a Swiss guy in the semifinal. That's positive," Federer said. "I will pull out all the stops to advance to the next round."
Before taking on Federer, Monfils was pushed twice to five-set matches, and he played with a stuffed nosed after catching a cold earlier in the tournament.
"Yesterday I was not good at all. Better than today, but had a bad night," he said. "I woke up repeatedly during the night. I didn't have any fever this morning, but my body was sluggish. I felt slow. I couldn't keep up the pace. So it was even harder. Much more complicated for me today."
Monfils blamed himself for getting sick and promised to be more careful with his health in the future.
"The best players, they are rarely sick," he said. "They are rarely injured. I'm talking about the top-10 players. Top-10 players, they are fit. They are not sick. The Big Four, they are always healthy. In a Grand Slam tournament they make sure that they are healthy for two weeks, and that's essential."