The Swiss second seed saved match points in the fourth set, and eventually prevailed 4-6 3-6 6-4 7-5 6-2 on Arthur Ashe Stadium.
It was just the seventh time in a major that Federer had successfully arrested a two-set deficit.
French 20th seed Monfils had not lost a set heading into his quarter-final with 17-time major winner Federer, and was still on a perfect run when he claimed the first two sets within one hour 18 minutes of the first ball being hit.
Federer claimed the third after getting the decisive break in the fifth game, and maintaining his advantage by dropping just two points on serve in his next three service games.
Monfils' chance to seal the upset came in the 10th game of the fourth set, as he opened up a 15-40 lead on the Federer serve - and two match points - but the Swiss saved the first when Monfils put a pass attempt long and the second with a brutal forehand down-the-line winner.
The game was the first of five straight for the five-time champion at Flushing Meadows, with Monfils - serving at deuce in the 11th game - double-faulting twice in succession to hand his opponent an easy break.
Federer took it to a fifth and opened up a break lead for 2-0 in the decider, as he raced towards the finish line.
When Monfils was broken to love in the fifth game, his spirit was virtually crushed as Federer went on to complete a brilliant comeback in three hours 20 minutes.
There was not much between the pair statistically, with Federer edging Monfils in the winners' count 48-43 and conceding slightly fewer unforced errors 44-49.
Federer will take on Croatian 14th seed Marin Cilic in the semi-finals.