The 31-year-old will face either world number one and two-time champion Rafael Nadal or 2009 winner Juan Martin del Potro in Sunday’s title match.
Anderson, the Johannesburg-born, Florida-based player will be taking part in his maiden major final, where he’ll attempt to become his country’s first Slam champion since Johan Kriek won the 1981 Australian Open.
Cliff Drysdale was the last South African man in the US final in 1965 but he was defeated by Manuel Santana.
“It has been a long road. This means the world to me,” said Anderson, whose career was at a crossroads in January when hip problems forced him out of the Australian Open and his ranking slumped to 80.
“This is why we work so hard. It was tough for Pablo and me. I was pretty nervous with it being the first time for both of us on the sport’s most famous stage.
“I’m over the moon. I’m happy to have another shot,” added the 32nd-ranked Anderson.
Carreno Busta had not dropped a set at the tournament, helped by playing four qualifiers, and he was a set to the good on Friday, breaking in the seventh game.
The Spaniard had pocketed the opener hitting just two winners and only one unforced error.
But that was sufficient as 6ft 8in (2.03m) Anderson committed 14 unforced errors as he dropped serve for just the fourth time in the championships.
With Hollywood star Robert Redford and Microsoft founder Bill Gates looking on, the big South African carved out the first break of the second set for 3-1 but handed it back straightaway in the next game.
Anderson was soon 5-4 ahead, hitting his 100th ace of the tournament in the process before leveling the tie in the 12th game courtesy of a Carreno Busta double fault and a fierce backhand winner.
He maintained his grip in the fourth set with breaks in the fourth and eighth games and as 12th seed Carreno Busta wilted, he broke again for 3-2 in the fourth in what proved the final blow.
Anderson finished with a power-packed 22 aces and 58 winners.
– Anderson winless versus foe -Anderson will have his work cut out if he’s to lift the trophy — he is 0-4 against Nadal and 0-6 facing Del Potro.
Nadal will tackle Del Potro, wary of the Argentine giant whose game he describes as having the power to leave opponents “dead.”
World number one Nadal, the 2010 and 2013 champion, enjoys an 8-5 career edge over Del Potro, who knocked out Roger Federer in the quarter-finals.
The 31-year-old Spaniard is 22-3 in Grand Slam semi-finals, winning the last 14 in which he has appeared.
However, his last loss at this stage came on the same New York hard courts in 2009 at the hands of Del Potro, who went on to shock five-time champion Federer in the final.
“When Juan Martin is playing well, it’s difficult to stop him. If you let him play from good positions with his forehand, you are dead,” said Nadal, who has lost his last two meetings on hardcourts with the 28-year-old.
While French Open champion Nadal looks to reach a 23rd Grand Slam final and claim a 16th title at the majors, Del Potro is making up for lost time.
Playing in his first semi-final at the Slams since losing the longest ever last-four clash to Novak Djokovic at Wimbledon in 2013, Del Potro is riding a tidal wave of emotion in New York.
Unable to prosper fully from his 2009 US Open breakthrough, when he was just 20, long-term struggles with wrist problems began that required four surgeries and pushed him to the brink of retirement.
Del Potro has been virtually running on empty. He had to save two match points and come back from two sets down to beat sixth seed Dominic Thiem in the last 16.
He almost retired from that titanic clash, suffering from breathing difficulties and a fever. DM
"Joyfully to the breeze royal Odysseus spread his sail and with his rudder skillfully he steered." ~ Homer