Sinner and Alcaraz served up a five-set thriller, with the Spaniard winning 2-6, 6-3, 3-6, 6-4, 6-3.
While the Italian isn't quite ready to put their rivalry alongside that of Roger Federer, Rafael Nadal and Novak Djokovic just yet, he and Alcaraz are sure to continue battling for the top prizes for some time.
"Well, you cannot compare us yet with the other players we had or still have," Sinner told reporters.
"But for sure, I mean, if we watch the results, we always, or most likely, when we play against, it's in the semis or final or quarters. Very important matches."
After the semi-final win, Alcaraz now takes a 5-4 lead in the head-to-head between the pair, and Sinner will be looking to put them back on level terms when they next meet.
"I think that's exciting for the game, especially when head-to-head is quite close," Sinner said.
"And the winner is happy and then the loser tries to find a way to beat him the next time, no? I think that's exciting.
"That's what I will try to do."
That familiarity can cause its own problems, and it also depends on which competition they meet in.
"You can see a little bit of tension sometimes of both players and both sides just because we know each other slowly a little bit better," Sinner said.
"Each time when we play against we expect a couple of things, and then, to mix up the plans, you make different choices sometimes on the court.
"I think, next time obviously is going to be different. Let's see on what surface we play on. Also, that's very important. Best-of-three or best-of-five is also different. You know, Grand Slam, there is a different approach."
Sinner hopes to return to Paris for the Olympics, and while he was worried after suffering with tension and cramps during Friday's loss, he believes his recent hip problems weren't an issue.
"No, the hip was okay. The match, the more it goes on, the right hip doesn't have the strength of the left hip," the Italian said.
"It's normal at this moment, so sometimes I feel a little bit, especially after two-and-a-half hours and until the four hours, but this is no excuse. I was moving good. I was feeling quite good on the court.
"So I'm not anymore worried about the hip. Just disappointed about how it went today. Just have to accept it."
Sinner will take over as world number one next week, and is now ready to begin preparations for the grass season, with Wimbledon coming up in July, where he reached the semi-final last year.
"I will work on the physical stuff, for sure. Maybe couple of days off. Then physical stuff," he said.
"And that's it. I mean, you know, grass season is different. The preparation is different. You use different muscles.
"I'm looking forward to it. I mean, last year I had a great Wimbledon, so let's see what I can do this year."
