The Court One crowd held their breath as a wicked forehand opened up match point for Anderson and he completed his first victory over the eight-time champion with a triumphant service.
Seven times Anderson was required to hold serve to stay alive and each time he was equal to the task.
The setting was not the only thing alien to the defending champion.
He made the first statement of intent by cashing on two backhand errors by Federer and breaking the Swiss' serve in the second game of the second set. But that turned out to be the least of his worries. He's absolutely rolling, and he's conserving energy, as his longest match was an hour and 45 minutes against Mannarino.
But he seemed to answer any question about whether he'd be addled by the unfamiliar environs or Anderson's massive serve in the first 26 minutes, closing the opening set with the impeccable form he'd shown in winning his first five matches without losing a set or his serve.
But it was tempting to see this as an aberration with Federer bouncing back immediately to level the game and then the tie-breaker, even if narrowly at 7-5 after having built a comfortable 6-3 lead that he squandered by extravagant over-hitting.
When Anderson broke again in the fourth, Federer was suddenly on the ropes. The South African was just a better player and better competitor on the day: he hit 28 aces, 65 winners and 31 unforced errors compared to Federer's 16 aces, 61 winners and 33 unforced errors.
The only other times he has surrendered a two-set lead came against Lleyton Hewitt in Switzerland's 2003 Davis Cup semi-final loss to Australia in Melbourne, to David Nalbandian in the 2005 Masters Cup decider.
On Wednesday, they lost 5-7, 6-4, 6-2 to Xu Yifan of China and Gabriela Dabrowski of Canada.
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Nishikori was bidding to become the first Japanese man to make the semi-finals in 85 years.
After two years of physical and mental struggles, Djokovic appears fit and confident and ready to try to add to his tally of 12 grand slam titles.
"Down two sets to love I tried my best to keep fighting". So far the four matches have been really, really good for me'. I don't think this one naturally you can pinpoint exactly, except my match points. Djokovic, a 12-time major champion, had not reached the final four of a Grand Slam event since the 2016 US Open.
Djokovic got past No. 24 Kei Nishikori 6-3, 3-6, 6-2, 6-2 in the quarterfinals.
Federer and Nadal are favourites to meet in Sunday's final - 10 years after their epic 2008 showdown - but Wilander believes Del Potro could throw a spanner in the works.
Later in the match, Djokovic was clearly frustrated and complained of "double standards" when Nishikori was not punished for a racket throw.
"I think it's not fair, but it is the way it is in the midst of the emotions of a match".
Del Potro said: 'I think I will be in good condition.
He then let rip at Ramos for failing to sanction Nishikori for a similar act in the fourth set.
"I think the grass is ideal for Djokovic this year, it helps his backhand", he said.