Rune outclassed Alcaraz on serve to win the opening set 6-3 and held firm in the second when the Spaniard stepped up his level before retiring four points into the tiebreak, having received medical treatment for an abdominal strain at 6-5.
Alcaraz will have nine days to recover for the November 13-20 ATP Finals in Turin, where he is set to make his debut.
"I thought it was a great match. (It's) unlucky for him," said Rune, who was leading 6-3 6-6 (3-1) and reached his first ATP Masters 1000 semi-final.
"I cannot stretch. I couldn't serve well, I couldn't hit the forehand well'' said Alcaraz, who withdrew with Rune leading 6-3 6-6 (3-1) in the quarter-final. "I preferred to retire and see it, and take care about it.
"Right now I have (to undergo) some tests on how it's going to be before Turin. I'm focused on trying to get better in the abdominal and trying to be at 100% in Turin."
Auger-Aliassime raced to a 6-1 6-4 win over Frances Tiafoe (24) to remain on track for a fourth ATP title in as many weeks, having won in Florence, Antwerp and Basel during his unbeaten run.
The Canadian cruised through the first set but Tiafoe dug deep to hold serve and save five match points before Auger-Aliassime sealed a 93-minute victory on his sixth attempt.
"I feel better and better," Auger-Aliassime said. "Of course right now I'm playing well and feeling good, but I see things that I can still do better and still improve on to be able to top the game."
Reigning champion Novak Djokovic (35) will take on Italy's Lorenzo Musetti (20) while Tommy Paul (25) will be up against Stefanos TsitsipasStefanos Tsitsipas in the other quarter-finals later on Friday.
