Sinner dispatched Japan's Taro Daniel 6-1, 6-4, while earlier in the day Alcaraz made short work of China's Shang Juncheng, winning 6-2, 6-2.
Sinner has said he is not in a "comfortable" situation thanks to a World Anti-Doping Agency (WADA) appeal against a decision by tennis authorities to clear him of wrongdoing after he twice tested positive for a steroid in March.
His recent blistering winning streak was broken by Alcaraz in the China Open final on Wednesday.
But there was no tiredness on display from either player on Saturday, with 23-year-old Sinner looking completely unruffled in the first set against the 93rd-ranked Daniel.
Daniel fell behind again in the second and despite rallying halfway through to gently test Sinner, the Italian kept his cool, ending the set 6-4.
"I had only one practice session yesterday but I felt very comfortable on the court," he said after the match.
"Today's performance means a lot to me."
Sinner will face Argentina's Tomas Martin Etcheverry on Sunday.
"It will be a tough one tomorrow, very physical, because me and Tomas know each other quite well now," he said.
'Really proud'
Earlier, arch-rival Alcaraz also looked comfortable throughout, breaking in the first game.
Teenager Shang became only the second Chinese player to win an ATP Tour singles title at the Chengdu Open last month, but he was unable to make much headway against the world number two.
Despite their enthusiasm for Shang, the crowd couldn't resist Alcaraz's charm.
At the Spaniard's encouragement, they broke into loud cheers when he won a particularly exciting rally in which he hit a tricky backwards shot to keep the point alive.
"I've been playing really good tennis lately so I want to just keep going and keep feeling it, keep feeling this good," 21-year-old Alcaraz said after the match.
"I just had one practice then was straight into this match, so to be able to show this level in the first match, I'm just really proud," he added.
Alcaraz will next face another Chinese player, wildcard Wu Yibing, who beat 25th seed Nicolas Jarry in the second round.
Rublev loses
The tournament lost its second top-ten player on Saturday, as Andrey Rublev fell to 19-year-old Czech Jakub Mensik, joining Norwegian Casper Ruud on the notable casualties list.
The Russian, a finalist at last year's Shanghai Masters, roared in jubilation after winning a tiebreaker to take the first set.
But 65th-ranked Mensik broke Rublev in the first game of the second set, and again in the fifth to draw even.
The Czech then came from behind in the third to beat the world number six for the second time this year, with a final score of 6-7 (7/9), 6-4, 6-3.
