Alcaraz beat twice in the first set against de Minaur before recovering and making a decisive breakthrough in Game 12.
The 21-year-old was even more dominant in the second set and received his only break in Game 6.
“I started serving terrible and I was struggling mentally,” Alkaraz said.
“It's just the first few games I served and then I think I've made a lot of progress. I calmed down and started thinking positively again. Alex started to play strong, actively and was really nervous in the first set.
“I’m really glad I managed to win the first set and in the second set I played some great tennis and was very happy.”
Four-time runner-up Tsitipas looked uncomfortable in the match against FILS and had an outside assessment at the end of the second game.
He returned, but could only manage 5 more points and retire, while 0-2 30-40 fell.
The 13th-ranked Rune broke the defending champion and second seed Rude twice in each set to win 6-4 6-2.
The sixth seed runes will play Karen Khachanov in the semifinals after Russia beat Spain's Alejandro Davidovich Fokina 6-4 7-5.
