He saved a match point at 4-5 when Federer missed a forehand, then broke to serve for the third set.
He clawed his way from two-sets down to win 2-6, 6-7 (5/7), 7-5, 6-4, 13-11 in four hours and 13 minutes.
The 36-year-old is also on a streak of 32 consecutive sets won at the tournament, just two off his record.
The three-time Wimbledon champion was broken in the opening game of the fourth set, but won the next four games and broke again to clinch victory.
During his Wimbledon fourth-round victory over Adrian Mannarino on Monday, with the ball having already been called out, Federer played a stroke that was more akin to cricket than tennis.
Beating Roger Federer here at Wimbledon will be one I remember, especially in such a close match.
Defending champion Roger Federer suffered the most shocking defeat of the tournament so far as he was knocked out by big-serving South African Kevin Anderson. Anderson and Federer are knotted at 3-3 in the final set. Those are the kind of circumstances that you just have to accept.' He also revealed: 'Post-match I heard that the organisers were planning to cancel my match if Monfils and Anderson went to a fifth.Читайте также: Klopp talks about Loris Karius after pre-season friendly
While Anderson is known for his booming serve, it's Federer's serve that has proved impenetrable at this year's Championships - the Swiss is yet to drop serve and has held in his past 81 service games, dating back to the first set of his semi-final against Czech Tomas Berdych last year.
"I just kept on telling myself, 'I have to keep believing.' I kept saying that today was going to be my day, because you really need that mindset taking the court against somebody like Roger", Anderson said.
It will be Djokovic's first semi-final at a major since the 2016 French Open when he completed the career Grand Slam.
"I don't think it's something anybody aims for, to win every match in straight sets". I had my chances and blew them, so.
He forced a tiebreak but, despite losing the first mini-break, Federer turned things around to go two sets ahead.
However since then, the 2.03m-tall Anderson has improved markedly and Wednesday's match will be their first meeting on grass.
The duo had met each other at the All England Club in 2011, when Federer was chasing his seventh Wimbledon title, and fans across the globe had gone into a frenzy seeing the legends in one frame.При любом использовании материалов сайта и дочерних проектов, гиперссылка на обязательна.
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