Defending champion Alcaraz had to dig deep to beat the in-form Italian, who came into the contest undefeated this season and on a 19-match winning streak dating back to last year.
Alcaraz and Australian Open champion Sinner are now 4-4 against each other.
"This makes things really fun, no?" Sinner told reporters.
"Maybe there is one day where one of us wins three, four times in a row. Then the opponent or the other one has to try to adjust a little bit, trying completely new things.
"Maybe goes completely wrong, that you lose 6-1 6-1, or whatever. But I think that's good to see for me and then trying to grow in the future."
The men's game has been searching for new rivalries since the retirement of Roger Federer in 2022 and the expected exit of Rafa Nadal sometime this year, which will leave just Novak Djokovic from the Big Three era.
Those concerns have been ameliorated by the ascent of world number three Sinner and two-time Grand Slam champion Alcaraz.
Sinner said he enjoyed his time on centre court in the California desert despite falling 1-6, 6-3, 6-2 in a match that featured several jaw-dropping rallies at the tournament many consider the most important outside the four majors.
"This is one of the best places to play tennis, no?" Sinner said. "Packed stadium. Two young players trying their best. This is fun to play there.
"For sure you want to win, but there will be occasions in the future where you win some and you lose some. There is nothing you can do about.
"Most important part is that you give 100%, which I've done today."
