"I think the first warning was unnecessary", said Djokovic, who was sanctioned in the second set for spearing his racquet into the court.
Anderson said after the match that although he was nearly down and out, he knew that he had to keep trying to get back into the match.
Anderson broke into the top-10 scene for the first time in 2015.
Eventually though, Djokovic pulled away with emphatic wins in both the third and fourth sets.
"I felt great in practice, good in the warm-up".
This was not vintage Federer, even in the first set which he won handily. Against someone like Roger, if you go out there with doubts, like I did in the first set, it's not going to go well.
It was an extraordinary turnaround after Federer won the first set in 26 scintillating minutes against a player he had beaten four times out of four without dropping a set. Still the focus was on the big three in the men's draw - Federer, Novak Djokovic and Rafa Nadal - but Anderson believed he could surprise many with victory, constantly talking to himself throughout the game.
There's nothing really that shocked me because I've seen Kevin play many, many times in the past. And he had held serve 81 games in a row at the grass-court major, a streak that grew to 85 before Anderson broke him a surprising total of four times.
Rafael Nadal edges del Potro to set Novak Djokovic clash — ATP Wimbledon
The Spaniard will face Novak Djokovic in the last four for a chance to meet either John Isner or Kevin Anderson in the final. The fifth seed had led by two sets on Monday but Simon grabbed the third set before fading light ended play.
With the scores deadlocked at 11-11 in the final set, and no obvious break in serve forthcoming, Anderson profited as Federer started to show signs of strain. I gave it my all.
The eight-time champion and 20-time grand slam victor last played a Wimbledon match outside Centre Court in 2015.
About five hours later, Federer's rival and foil, second-seeded Rafael Nadal, survived a five-set spectacle against fifth-seeded Juan Martín del Potro, 7-5, 6-7 (9-11), 4-6, 6-4, 6-4.
As it stands, it looks like it very well could be a repeat of the classic 2008 Wimbledon final where Nadal and Federer face each other in the final and rekindle memories of what is arguably the greatest tennis match of all time. The South African made no mistake.
But, playing on Court One for the first time in three years, Federer was unusually error-prone. Then in the 23rd game, at 11-all, 30-30 Federer blinked, sending down a double-fault and then drove a forehand into the net to allow his giant opponent to serve for a place in the semifinals.
"Down two sets to love I just tried my best to keep fighting and scraped through the third and fourth set", Anderson said. Of course, the goal is to come back here next year. "It's not like I would have remembered exactly how it would have felt", he said.
'I'm pleased I'll be staying there right now'.
It was Federer who faltered at 11-11, double-faulting to hand Anderson a break point which he converted when the defending champion's tired forehand smacked the net.