Alcaraz twice fell a break down in the first set against De Minaur before recovering and making the decisive break in the 12th game.
The 21-year-old was more dominant with his serve in the second set and made the only break in the sixth game.
“I started quite badly with the serve, I was struggling a little bit mentally with it,” said Alcaraz.
“It was just the first few games with my serve, then I think I improved a lot. I calmed myself down and started to think positively again. Alex started strong, playing aggressively, and the first set was really tight.
“I’m really glad I managed to win the first set and in the second I played some great tennis, so really happy.”
Four-time runner-up Tsitipas looked uncomfortable from the off against Fils and left for an off-court assessment at the end of the second game.
He returned but could only manage five more points and retired while 0-2 30-40 down.
World number 13 Rune broke defending champion and second seed Ruud twice in each set to secure a 6-4 6-2 win.
Sixth seed Rune will play Karen Khachanov in the semi-finals after the Russian beat Spain’s Alejandro Davidovich Fokina 6-4 7-5.
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