The flamboyant Frenchman clobbered a below-par Federer 6-1, 6-4, 6-3 in just one hour, 46 minutes.
It was just the second time that Federer had lost a Davis Cup singles rubber in straight sets — the other coming in his 1999 debut against Gianluca Pozzi of Italy.
Earlier Stan Wawrinka had given the Swiss favourites a 1-0 lead by outplaying French No 1 Jo-Wilfried Tsonga 6-1, 3-6, 6-3, 6-2.
Wawrinka's win came as an all-time Davis Cup record crowd of 27,432, crammed into one half of a roofed Lille football stadium.
Tsonga looked and played tight from the start but in the latter part of the match his big serve deserted him after he won the second set.
Federer took a 8-2 winning head-to head record against Monfils into the rubber.
But his preparations for his first ever Davis Cup final took a hit last Saturday when he injured his back playing, ironically against Wawrinka, in the last four of the ATP World Tour finals in London.
He subsequently pulled out of the final against Novak Djokovic and was unable to practice on the indoor claycourt at Lille football stadium until Wednesday evening.
The French, who had no players qualified for London, had two weeks training on clay in Bordeaux and it showed as Monfils opened with an ace and had Federer on the back foot from the start.
A superb backhand passing shot brought up break point for Monfils in the fourth game and he then forced a backhand return into the next to move 3-1 up.
Federer looked slow and uncomfortable, especially on the backhand flank, and he surrendered his serve again two games later as the first set slipped away from him in just 32 minutes.
The rubber was unwinding at breakneck speed and it was a poor Federer forehand into the bottom of the net that gave Monfils the first break of the second set.
The way the Parisian was serving, with nine aces up to that point, made it a steep hill to climb for Federer and he made no further inroads as he dropped the second set 6-4.
The Swiss star, 37-7 in Davis Cup singles, desperately needed to get his nose in front to try and put some pressure on the Monfils serve.
He did manage to hold his first two serves in the third set, but Monfils held firm and then the Frenchman struck what proved to be a fatal blow in the fifth game.
A typically sliding backhand passing shot set up break point, which was promptly secured when a dispirited looking Federer slapped a forehand well wide.
There was no way back for the 17-time Grand Slam title winner, who is playing in his first Davis Cup final at he age of 33.
Monfils clinched a famous and surprisingly easy win with a typically flashy backhand drive down the line from well behind the baseline.
It was the first time that Monfils had beaten Federer on clay and it could not have come at a more opportune moment.
Saturday's doubles sees the French pair of Julien Benneteau and Richard Gasquet go up against Marco Chiudinelli and Michael Lammer, with the reverse singles on Sunday.
Given the low-key showing he produced on Friday, the doubts will persist that Federer may not be able to complete the tie which would leave the Swiss in all sorts of trouble.