However, he later handed Djokovic another break with a wide backhand, allowing Djokovic to serve out and take a comfortable first set.
Although he was broken twice in the opening set, Djokovic was rattling through the games at supersonic speed. He was within touching distance as he held to 30 for a 5-2 buffer and, much like the preceding set, he rammed home his superiority to seal the set - and the match - with a double break.
If Anderson is to shock Federer, who has only lost once before the quarter-finals at Wimbledon in 15 years, he will need to call up the heavy artillery which has served him well at the tournament this year.
Nishikori, playing Djokovic on grass for the first time and seeking to end a 12-match losing streak against the Serb, held his own in the third set before running out of steam.
Three times champion Novak Djokovic roared back after dropping the second set to beat Japan's Kei Nishikori 6-3 3-6 6-2 6-2 and reach his eighth Wimbledon semi-final. It will be the 32nd Grand Slam semi of his career and first since the 2016 U.S. Open but, given what has happened since his previous one, a special feeling nonetheless.
But the 31-year-old wasn't happy with Carlos Ramos, accusing the the Spanish official of "double standards" during an ongoing battle in the two-hour, 35-minute encounter.
'I thought it was unnecessary to get the warning, ' he told the BBC after the 6-3 3-6 6-2 6-2 win. "It didn't harm the grass". (Ramos) said he didn't see it. I think his confidence is back, ' he said.
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"It took me two or three months and that was frustrating that I wasn't able to perform the game that I was used to and the level of tennis I wanted to play".
A similar situation arose in fifth game of the third set but this time it was the Serbian who saved three break points before he let out a huge roar as the momentum was on his side.
"You think I ruined the court for just throwing my racket (down)?"
Defending champion Federer lost a Court One thriller, 2-6, 6-7 (5-7), 7-5, 6-4, 13-11 as 32-year-old Anderson became the first South African in the Wimbledon semifinals since Kevin Curren in 1983.
The semi-finalist went into more detail about his issues after the match. "I knew how I threw the racquet", he said.
After breaking Djokovic's first serve to get off to a promising start in the fourth set, Nishikori was immediately relinquished of his lead as Djokovic broke back once and then again. But the 12th seed was not to be denied as he raced into the last four, hitting an impressive 40 winners on the way.