Second-seeded Alcaraz, playing his first tournament since he was sidelined by a right wrist injury in April, looked closer to his brilliant best in a 6-4, 6-3, 6-4 victory over Paul.
The 20th-seeded American posted the sternest test yet for Alcaraz, and he passed it with flying colors.
“I played my game,” said Alcaraz, who unleashed 33 winners as he moved fluidly around the court in the two-hour 20-minute match.
“I tried to play my style,” he said.
“Tommy is such a talented player. His shots are pretty dangerous and sometimes it’s really difficult to control the ball when it comes to you.
“I just tried to be ready for a battle and tried to be aggressive all the time.”
Alcaraz grabbed the first break of the contest in the final game of a tight first set.
From there, it was almost all Alcaraz.
He broke Paul twice in the second set and again early in the third.
He finished it off in style as he saved a pair of break points in the final game, the second with a swinging forehand volley followed by a forehand winner to set up the only match point he needed.
“I’m really happy with the performance today,” said Alcaraz, who next faces either in-form eighth-seeded American Ben Shelton or resurgent Greek Stefanos Tsitsipas.
Sabalenka kept her bid for a rare US Open three-peat on track with a 6-4, 6-3 victory over American Taylor Townsend.
The Belarusian, whose World No. 1 ranking is on the line in New York, will face a tough quarter-final against sixth-ranked Czech Noskova, who edged dangerous Ukrainian Marta Kostyuk 7-5, 5-7, 6-4.
Sabalenka needed all of her resolve against the multi-faceted Townsend, whose blistering service returns and net prowess flummoxed the Belarusian star in a first set that opened with five breaks of serve.
Sabalenka managed a comfortable hold for a 4-2 lead and eventually her sustained power and precision proved a little too much.
“What an incredible player she is, approaching the net, what a great returner, moving well, very brave,” Sabalenka said.
“You can see how focused she was and it was really challenging playing her, especially here.”
The second set unfolded in more traditional style, neither player facing a break point through the first three games. (AFP)