Federer was on the ropes after dropping a comfortable two-set lead, and he even failed to take a match point chance in the third set, but the South African held his nerve to win a captivating contest 2-6 6-7 (5-7) 7-5 6-4 13-11.
Now, as the eighth ranked singles player on the ATP World Tour, Anderson is a force to be reckoned with at the professional level.
No tournament wants to see Federer go out and we'll be denied a repeat of that 2008 Federer v Nadal final, which is a shame.
"Obviously I'm very ecstatic to get through that match", Anderson said.
"I felt good actually".
"Rafa [Nadal] and Roger played all four matches on Centre, myself only one". Maybe I don't have a good record with him, but I always enjoy playing him'.
It was only the fifth time in Federer's career that he lost a best-of-five match that he led, two sets to none. Nadal will advance to play in his sixth Wimbledon semifinal, and his first since losing in the finals in 2011.
He is a bit of a sporting contradiction - a quietly spoken man, with the loudest of bombastic serves.
The escape energised Anderson and he broke in the next game with a bludgeoning backhand off a weak second serve and then battled back from 0-40 to seal the third with an ace. He could potentially win a set and if he can win a set he could win two sets.
He won the third set to snap Federer's 34-set winning streak at Wimbledon and Federer was clearly shaken as Anderson dominated the fourth set to drag the 20-time Grand Slam champion into a decider.
Shark Attacks Model During Instagram Photoshoot
The nursing student recalled everything being fine when she first entered the water - until she leaned back and held her arms out. The views expressed therein are not necessarily those of stlucianewsonline.com, its sponsors or advertisers.
For so long it had looked like Federer's championship to lose.
Federer had breezed into the quarter-finals and was imperious in the opening set. Few considered Anderson as having a chance. He wastes three set points and gives one to Anderson, instead. And here at Wimbledon, there has been a quiet confidence about Anderson.
The 12-time major victor prevailed despite picking up two code violations and accusing umpire Carlos Ramos of "double standards".
Like Federer and Anderson, Nadal and del Potro Nadal slugged it out for more than four pulsating hours before Spain's 17-times major victor finally landed the knockout punch.
"To be honest, I didn't feel mental fatigue (during the match)".
Anderson will now rest ahead of Friday's final-four match. "It feels great to make the last four of a Slam again". Some of that was because the South African continued to attack at every opportunity.
"Yeah, of course, the goal is to come back here next year", he said.
"Roger is playing extremely well, moving well".
"That was something I learned from 2016 where I constantly tried to get back into competition too quickly". After knocking out Monfils in Round 4, he spoke of the need to approach this match on his own terms.