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, and 0-40 on his own serve, to force a tiebreak in the opening set before the Spaniard came out on top.
The pair traded breaks at the start of the second set, but Alcaraz raced into another 4-1 lead, and this time there was no comeback despite De Minaur's valiant effort in the opening match of the Jimmy Connors Group.
Alcaraz had lost the opening match at his previous two appearances, and while his aggressive approach led to 19 unforced errors in Sunday's first set, he never let up and the win was never in doubt once he took charge in the second set.
"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."
After winning the tiebreak, Alcaraz began to enjoy himself a little more, cupping his ear to milk the applause that greeted him winning a long, entertaining rally to hold serve and lead 5-1, and there was no way back for De Minaur.
Alcaraz is also aiming to end the year as world number one for the second time in his career, and needs to reach the final with at least one round-robin win or sweep all three of his round-robin matches.
Italian Jannik Sinner needs to retain his title to have any chance of finishing the year in top spot.
One of the loudest cheers in the arena came when an advertisement featuring the Wimbledon champion was shown on the large screens, but the crowd needed to wait until Monday to see the home favourite in action.
Zverev off to flying start
American Shelton, making his first ATP Finals appearance, looked set to force a third set but threw away a tiebreak that appeared all but won and Zverev came through 6-3, 7-6(6) in their Bjorn Borg Group clash.
Shelton was overpowered by Zverev's serve in the opening set while the German, who won three service games to love, broke the American twice to take the opener in 28 minutes.
The American saved two break points to avoid going 3-1 down in the second, and hung in to force a tiebreak, where he went 4-0 ahead. Shelton was 6-3 up and serving for the set when Zverev made a stunning comeback with three successive breaks of serve.
The ATP Finals 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.