"I'm not quite sure what to say right now but down two sets, I really just tried my best to keep fighting," Anderson told the BBC in his post-match interview. "I was able to scrape through that third set and the fourth set and by the end I thought I did a great job of not thinking about things too much. I felt I was really in the flow of the match.
"Beating Roger Federer here at Wimbledon will definitely be one that I remember, especially in such a close match."
Federer won the first two sets of the match, before Anderson claimed sets three and four. The South African won the quarterfinal matchup by upending the eight-time tournament champ in a 90-minute fifth set. The final scoreline read in 2-6, 6-7, 7-5, 6-4, 13-11 in favor of the eighth-seeded Anderson.
Anderson had 28 aces in the victory, compared to Federer's 16 aces. Anderson hit 11 of those winners in the fifth set.
Federer -- the No. 1 player in the ATP world rankings -- was previously 4-0 against the South African. Anderson moves on to face Milos Raonic or American John Isner in the semifinals on Friday at the All England Club.
"I guess there was definitely a moment at some point (I lost control of the match)," Federer told reporters. "Is it missing match points? Is it getting broken? ... So that could have been a key. It could have been a key later. There are a lot of little points here and there that always make a difference in the outcome of a match."
No. 12 Novak Djokovic also advanced to the semifinals. The Serbian will face No. 2 Rafael Nadal or Juan Martin del Porto in the semifinals in London. Djokovic beat Kei Nishikori 6-3, 3-6, 6-2, 6-2 in his quarterfinal matchup.
"He had to play a great, great game to break me," Federer said of Anderson. "If that's what it takes, that's alright."