The 19-year-old dropped just two games as she blitzed Jelena Ostapenko 6-0 6-2 inside Arthur Ashe Stadium.
Latvian Ostapenko, the 20th seed, had blown the women’s draw wide open when she knocked out defending champion and world number one Iga Swiatek in round four.
But she was unable to take advantage of her own handiwork as Gauff raced away with the first set in only 20 minutes.
Ostapenko made more of a contest of it in the second set, finally managing to hold serve at the fifth attempt.
But Gauff completed the job in an hour and eight minutes to reach the last four of her home grand slam for the first time.
“It feels great, I’m so happy,” she said. “Last year I lost in the quarter-finals and I wanted to do better. There’s a long way to go but I’m happy and ready to go back to work.
“Honestly, I didn’t feel comfortable the whole match, even on match point. I know the game she plays. It’s really tough against her – you can’t really be aggressive. She’s a top player and she’s had a great tournament.
“There’s a saying in basketball that ‘defence wins games’. It doesn’t always work in tennis but today that was the case.”
Gauff will face the winner of Tuesday night’s match between Sorana Cirstea and Karolina Muchova in the last four.
The defending champion had to dip into his armoury of explosive winners to finally see off the British number two inside Arthur Ashe Stadium.
The last time Evans played the top-ranked player he rose to the occasion, beating Novak Djokovic in Monte Carlo three years ago.
And the 26th seed brought the very best out of the 20-year-old Wimbledon champion before succumbing 6-2 6-3 4-6 6-3.
Alcaraz dominated the first set, constantly pushing his opponent to the back of the court and hitting 14 winners to Evans’ four.
But he showed he is not infallible at the start of the second after a double-fault gave Evans, who was beginning to relish the challenge, a break point which he converted with a superb backhand down the line.
Alcaraz quickly retrieved the break and a frustrated Evans was given a code violation after slicing the ball into the crowd after netting a volley at 15-30 up on the Spaniard’s serve.
Evans was playing well, but Alcaraz simply raised his level and brought up set point with a blistering forehand down the line which brought gasps from the spectators.
Evans saved that one, but could do nothing with the next after an exquisite lob and simple volley put Alcaraz two sets up.
Never one to give up without a fight, Evans promptly played one of the best sets of his career.
He broke for 4-3 and, serving for the set, forced Alcaraz into some of the most outrageous clutch winners.
The pair, who played golf together at Wimbledon, frequently exchanged knowing looks and grins, such was the quality of the cat-and-mouse tennis on show.
It took Evans, 33, five set points to make the breakthrough, prompting a suddenly exasperated Alcaraz to slam his racket down in frustration.
But Alcaraz turned the match back in his favour with another stunning shot on break point. On the run and at full stretch, he unleashed a forehand down the line which clipped the baseline and left Evans stood, hands on hips, in sheer disbelief.
Two aces, another sensational lob onto the baseline and a 61st winner of the match finished Evans off, but Alcaraz knew he had been in a contest.
“He’s a tricky opponent, great slices, good touch and it’s my game as well so it was a pretty good match,” said Alcaraz.
“We made great points, great shots, a lot of different situations, so I’m really happy to get through
“We played some great points that made us smile, Dan as well. It’s great to see Dan and I on a tennis court, we tried to entertain and make the match fun.”
Three people, wearing T-shirts bearing the words ‘fossil fuels’ began shouting after the first game of the second set, forcing play to be stopped.
American teenager Gauff and Czech 10th seed Muchova initially stayed on the court for about 10 minutes while security tried to deal with the situation.
The Arthur Ashe crowd at one point began chanting “kick them out” with the protesters apparently being difficult to shift.
“Are they like talking to them or are they going to remove them?” Gauff asked chair umpire Alison Hughes and tournament referee Jake Garner.
She then spoke to her coach, Brad Gilbert, saying “they say they are negotiating, like it’s a hostage situation. What should I do?”
Both players eventually left the court with Gauff leading 6-4 1-0.
The two-time winner, handed a wild card at the age of 43, was beaten 6-1 6-1 in the first round by Belgian Greet Minnen.
The American said: “I have to give credit to my opponent, there wasn’t a shot she couldn’t make.
“I don’t think I played badly, it was just one of those days where I was unlucky.
“I was really happy to be here. I love playing here and I really gave it my all today.”
Wimbledon runner-up Ons Jabeur had to overcome breathing difficulties to battle past Columbia’s Camila Osorio.
The Tunisian had her blood pressure checked at one point before securing a 7-5 7-6 (4) victory.
France’s Caroline Garcia, seeded seven, suffered a shock exit, 6-4 6-1, to world number 114 Wang Yafan of China.
Canadian Laylah Fernandez, the runner-up to Emma Raducanu two years ago, lost in three sets to Ekaterina Alexandrova.
