Having already played on Friday, defeating compatriot Jaume Munar 6-3 6-3, 18-year-old Spaniard Alcaraz went the distance against Monte Carlo champion Tsitsipas.
Showing his proficiency on clay, Alcaraz triumphed 6-4 5-7 6-2 to set up a semi-final against Australia's Alex de Minaur, who prevailed 6-3 5-7 6-1 over Cameron Norrie.
Alcaraz, the No.5 seed, was in excellent form against the No.1 seed and last year's runner-up, and cruised ahead in a dominant first set, striking 14 winners to Tsitsipas's five.
Indeed, Tsitsipas, who had beaten Grigor Dimitrov in straight sets earlier in the day, could not hide his frustration when he smashed the ball hard Alcaraz's way to give up the set, much to the anger of a partisan crowd.
Alcaraz seemed en route to having things wrapped up quickly as he followed up a supreme backhand winner with a sensational drop shot, yet from 4-1 down, Tsitsipas reeled off four straight games to nudge himself ahead, breaking to take the set.
That comeback took just too much out of Tsitsipas, though, with Alcaraz roaring into a 4-0 lead in the decider and, after fending off two breakpoints, sealed his biggest win so far at the first time of asking to reach a fourth semi-final of 2022.
Alcaraz has won all three of his meetings with the Greek and will break into the top 10 of the ATP rankings next week, becoming the youngest player to do so since Spanish great Rafael Nadal in 2005, after his first title in Barcelona.
Indeed, he will be the ninth youngest player overall to make the top 10 and the 20th teenager to do so, though the first since Andy Murray in 2007.
"Probably my biggest win on clay court. It was unbelievable," Alcaraz said.
"Unbelievable game that I played, unbelievable atmosphere that I lived today on court. It was unbelievable everything. The atmosphere here, the crowd, the level that I played, the level of the match. It was incredible.
"I'm playing an incredible level. And I think that I'm ready to get the title."
Vying for a place in the final in the other semi-final will be Diego Schwartzman and Pablo Carreno Busta, who both produced remarkable turnarounds in their quarter-final matches.
Schwartzman came back to beat Felix Auger-Aliassime 3-6 6-2 6-3, following on from an earlier win over Lorenzo Musetti.
Carreno Busta, meanwhile, toppled Casper Ruud 4-6 7-6 (10-8) 6-3, though only after saving three match points in the second set in a victory that took three hours.
At the Serbia Open, No.2 seed Andrey Rublev defeated qualifier Taro Daniel 6-3 6-3 to progress to the semi-finals.
In windy conditions in Belgrade, Rublev had little trouble in securing a 74-minute win, teeing up a clash with Fabio Fognini and moving to 21-5 for the season.
Rublev has met the Italian on nine occasions previously, losing five times and winning on four occasions.
"It's going to be tough. Fabi, he's super talented, he can play amazing and it's going to be a tough match with a lot of long rallies, so I just need to do my best and we will see what happens," Rublev said.
Fognini made light work of defeating Oscar Otte 7-5 6-4.