Alcaraz was on top for most of his 7-6 (7-1) 7-5 victory, winning 80 per cent of his first-serve points and 75 per cent of net points against Sinner.
Sinner showed determination to stay in both sets, with the Italian saving nine of the 11 break points he faced, but it was ultimately in vain.
The European Open winner will be among those sweating as the race to qualify for the ATP Finals in Turin heats up.
After the win, 18-year-old Alcaraz said: "I'm so happy for this win as Jannik was fighting for a spot at the ATP Finals. It's my third Top 10 win of the year.
"I think Jannik and I will have a great rivalry in the future... I think that I played really, really aggressive, more than him. I think that was one of the keys."
Alcaraz will now face qualifier Hugo Gaston in the next round after the Frenchman impressively knocked out 12th seed Pablo Carreno Busta 6-7 (3-7) 6-4 7-5.
Interesting day for Turin hopefuls
It was a mixed day for others looking to secure a spot at the season-ending ATP Finals later this month as Felix Auger-Aliassime – ranked 12th in the ATP Race to Turin – lost in straight sets to Dominik Koepfer, who added to his impressive list of victims after beating three-time grand slam champion Andy Murray in the first round.
Koepfer will play another Turin hopeful, the 10th-ranked Hubert Hurkacz after he beat Tommy Paul in straight sets.
Cameron Norrie also impressed in his 6-3 6-4 win over Reilly Opelka, which was his 50th tour-level win of the year.
Taylor Fritz awaits after the American stunned fifth seed Andrey Rublev 7-5 7-6 (7-2).
Medvedev and Zverev ease through but Tsitsipas out
Second seed and US Open champion Daniil Medvedev had a routine first match of the tournament as he swept past Ilya Ivashka 7-5 6-4, while Olympic Games gold medallist and fourth seed Alexander Zverev also had few problems against Dusan Lajovic 6-3 7-6 (7-5).
However, third seed Stefanos Tsitsipas exited after retiring hurt with an apparent arm injury against Alexei Popyrin in the first set with the score at 4-2 to the Australian.
Popyrin will now face fellow countryman James Duckworth, who followed up his impressive win against 14th seed Roberto Bautista Agut with a 6-3 3-6 6-3 victory against Lorenzo Musetti.
Elsewhere, 11th seed Diego Schwartzman was shocked by qualifier Marcos Giron 7-6 (7-2) 7-6 (7-4).
There were also wins for 16th seed Grigor Dimitrov against Karen Kachanov, and Sebastian Korda over Marin Cilic.
Gael Monfils will go up against world number one Novak Djokovic in the third round after the experienced French 15th seed came from behind to beat compatriot Adrian Mannarino 2-6 7-6 (7-4) 6-2.