Alcaraz won his opening ATP Finals match for the first time despite De Minaur's battling display, where the Australian recovered from 4-1 down to force a tiebreak in the opening set before the Spaniard came out on top.
The pair traded breaks of serve at the start of the second set, but Alcaraz broke twice more to again lead 4-1, and this time there was no comeback despite De Minaur's valiant effort in the opening match of the Jimmy Connors Group.
Both players were looking to improve on last year's showing, where Alcaraz lost two of his three round-robin matches in straight sets while De Minaur lost all three group matches, with Alcaraz losing the opening match at his previous two appearances.
"It was a really difficult match," said Alcaraz.
"On this surface, Alex makes the most of the speed of the ball, he's super fast and really difficult on return. I'm just really happy to get through this really difficult challenge."
Alcaraz is also on course to end the year as world number one, and in Turin, he must reach the final with at least one round-robin win or sweep his round-robin matches to guarantee himself the award for the second time.
Italian Jannik Sinner needs to retain his title to have any chance of finishing the year in top spot.
Later on Sunday, German two-time champion Alexander Zverev takes on American Ben Shelton, making his first ATP Finals appearance, in the Bjorn Borg Group.
The ATP Finals, which features the top eight singles players and doubles teams in the world, has a record prize pool of $15.5 million and offers 1,500 points. The tournament culminates with the final at the Inalpi Arena in Turin on November 16th.
