The 12th seed was sent tumbling out of the tournament by the big-serving South African in a four-set battle on Arthur Ashe, before Nadal followed him into the final after downing Juan Martin del Potro.
The momentum quickly changed in the second, however, a scorching forehand giving Anderson the break, only for Carreno Busta to hit straight back with a fine forehand down the line.
The ninth-seeded Williams meets No. 13 Petra Kvitova on Tuesday night in a quarterfinal matchup of former Grand Slam champions.
The likes of fourth seed Alexander Zverev, seventh seed Grigor Dimitrov and eighth seed Jo-Wilfried Tsonga then went out in the early rounds.
Carreno Busta displayed great resilience to save two break points at 4-3 down, but Anderson's persistence paid off in the final game of the set as he broke courtesy of angled backhand.
Not since Kevin Curren made the 1985 Wimbledon final - he was a joint US and South African citizen - has the country had someone making it this far at one of the sport's biggest tournaments, and he was supported by Major-winning golfers Charl Schwartzel and Louis Oosthuizen in the Arthur Ashe Stadium. But he was constantly under pressure on his serve, while Anderson used his power to pound 22 aces. But Anderson has far more career match wins with 267 compared to Carreno Busta's 110.
US Open: No. 1 seed Karolina Pliskova rolls into US Open quarters
She dropped the first set and had to fight off a match point in the second before rallying to beat No. 27 Zhang Shuai 3-6 7-5 6-4. Kanepi, who's missed nearly two years of play with plantar fasciitis, is playing her first Grand Slam since the 2015 US Open.
"We've been tampering with a few things, and this is the one that's locked in the most", Godwin said.
Anderson holds a 2-0 record against his opponent, Pablo Carreño Busta, who has yet to drop a set in the tournament. He committed 14 unforced errors to just one for Carreno Busta in the set. Or will the Spaniard continue his rich vein of form at Flushing Meadows and keep his dream alive? Anderson hasn't won an ATP title since 2015 and has won 28 and lost 15 matches this year.
Tecau and Rojer eliminated top-seeded Henri Kontinen and John Peers in the semifinals.
"Everybody knows Rafa, he's an incredible tennis player and one of the best ever", said Carreno Busta.
The matches Friday offer quite a contrast in experience and name recognition. I have given myself a shot at being in the finals and I will be playing for a grand slam trophy. Having won their previous two meetings, Keys went on to cruise past Vandeweghe 6-1 6-2 in a bit over an hour of play.
However, his last loss at this stage came on the same NY hard courts in 2009 at the hands of Del Potro, who went on to shock five-time champion Federer in the final.
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