Monfils had Federer tied in knots, but his nerve failed him just as he should have gone in for the kill - and once Federer had wriggled off the hook, he picked himself up in superb style to go on and win the match.
Monfils struck a backhand long to squander his first chance to claim victory and then Federer denied the 20th seed with a sizzling forehand winner to stay in the match.
From that point on, Federer took charge by holding serve for 5-5, closing out the fourth set and then breaking Monfils in the first game of the fifth.
Playing at his brilliant best after scattering unforced errors earlier in the match, the Swiss master took nine of the last 11 games in the three-hour 20-minute clash at Arthur Ashe Stadium.
The turnaround says a lot about the talented but enormously frustrating Monfils; but it says even more about Federer, who continues to defy his years - he's now 33 - despite having been written off as a major force in the game half a decade ago.
[MATCH REPORT: Federer comes back from two sets down to reach semi-finals]
HOW DOES HE DO IT?
Ahead of the US Open, the Daily Telegraph published a fascinating article discussing Federer's longevity - and putting it down to his natural balletic grace around the court.
The ATP Tour's lead analyst Craig O'Shannessy told the paper that Federer, "floats around so lightly, almost as if he’s skipping rope, that you can hardly hear him. He has very elastic ankles and great strength in his calves so that the lower leg does most of the work. He’s not carrying a huge amount of muscle – just the right amount to get the job done."
Andy Murray made a similar point on Wednesday night in New York after his clash against Novak Djokovic - pointing out how much easier on his body a Federer-style match is: "The way that we (Djokovic and I) play against each other, it's just an extremely physical match. Whereas, maybe when I play against Roger, for example, it's quicker points. So physically, that's not as demanding."
The Telegraph's correspondent Simon Briggs points out how Federer's approach is in stark contrast to Rafa Nadal, giving him the ability to contend at age 33: "Can you imagine Nadal, who is only 28, still contending for major titles at that sort of age? Hardly. It is a rarity for him just to make it through a year without some catastrophic breakdown. The last time he played all four majors in a season was in 2011.
"We can probably agree that Nadal at his best would beat a fit and firing Federer on the majority of surfaces. Even on grass, judging by the epoch-defining Wimbledon final of 2008. But could Nadal ever be as great, for as long, as Federer has been? Not a chance."
"When I was down two match points, I wasn't feeling so great ... I thought 'this is it, this is the last point, man. Just go down fighting, don't miss an easy shot and let him have it.'
"I served well and stayed in the match and somehow turned it around. I felt great in the fifth, though. I started playing better and better as the match went on. It's a great feeling.
"It was one of those moments where you got the back against the wall and hope to get a bit lucky and you hope to play exactly the right shots that you need or that he completely just messes it up.
"Either way works as long as you get out of it. It's just unbelievable to win matches like this at slams.
"I'm not sure I have ever saved match point before in a slam. If that hasn't happened, I'm unbelievably happy that it was today."
Despite Federer's brilliant performances and brilliant season, there is still a long way to go before Federer can even think about an 18th Grand Slam singles title. Marin Cilic was in devastating form as he brushed aside Tomas Berdych in Thursday night's other quarter-final, and will be a tough opponent.
Should Federer progress, many would expect Novak Djokovic to be a prohibitive favourite in the men's final - but that is not at all true, and Federer will rightly fancy his chances.
He leads Djokovic 18-17 head-to-head in their careers, took him to five sets at Wimbledon a couple of months ago, and beat him on a hard court in Dubai earlier this year. On top of that, all but one of their last six matches has gone to a deciding set- the only exception being the Monte Carlo Masters semi-final earlier this year, which Federer won in two sets.
The two faced each other at Flushing Meadows for five consecutive years from 2007 to 2011. Federer won the first three of those matches for the loss of just one set; Djokovic won the next two, but both were five-set classics. At least that much could be expected if 2014 witnesses their sixth US Open battle.
- Sports & Recreation
- Roger Federer
- Gael Monfils
- Novak Djokovic