The 18-year-old had to claw his way back from two sets to one down, eventually winning 5-7 6-1 5-7 6-2 6-0 to become the youngest man in the Open era to reach the US Open last eight.
Alcaraz will face another rising star, 21-year-old Félix Auger-Aliassime, for a place in the semifinals after the Canadian defeated Frances Tiafoe in four sets on Sunday.
"I'm super excited to be in my first second week in the grand slam, so it's amazing for me," Alcaraz said after the match. "It's a dream come true.
"It's really tough to play these kind of matches, to play fifth sets. I hope to play more second weeks, to play more quarterfinals of grand slams. I didn't expect to play quarterfinals here. So I think it's a really good performance from me in these matches."
That shock win over Tsitsipas -- which Alcaraz secured in a nail-biting, fifth-set tie break -- added to what has already been an incredible breakout season for the youngster.
Alcaraz won his first ATP Tour title in July, beating Richard Gasquet in the final of the Croatian Open, and also reached the third round at the French Open.
A teen revolution in Flushing Meadows
On the women's side of the draw, the teenagers are continuing to take over the US Open as 19-year-old Leylah Fernandez, who celebrates her birthday on Monday, came from a set down to beat former US Open champion Angelique Kerber 4-6 7-6 6-2.
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The Canadian teenager had already produced the shock of the tournament so far by beating defending champion Naomi Osaka in the previous round but followed up that win with another stunning performance.
Fernandez, who had never previously been past the third round at a grand slam, will now face Elina Svitolina for a place in the semifinals.
"I did have to pinch myself a little bit to see that it actually happened, but I knew that my tennis, my level of tennis is there," she said after the match.
"It's just a moment of time, opportunity, and I'm glad that I was able to get this opportunity now and that I was able to showcase what I can do in front of these players. Even if I would have lost, I would have been proud of myself with the way that I played, the way that I fought.
"I was lucky enough to get the win today and just get this experience. It's my first quarterfinals in a grand slam. I'm just going to enjoy it at 100% and tomorrow is going to be a new day."
Alcaraz and Fernandez could yet be joined in the quarterfinals by another teenager, with 18-year-old Briton Emma Raducanu set to face Shelby Rogers in the last 16 on Monday.
Raducanu burst onto the scene at Wimbledon earlier this year where she reached the fourth round and will now be aiming to go one better at Flushing Meadows.