Sinner, who fired down 10 aces at a rocking Arthur Ashe Stadium, became the only man to reach the last eight at all four majors this year.
He next plays 2021 champion Daniil Medvedev in a rematch of their Australian Open final this year, which the Italian won in a five-set thriller.
"Tough match. It's going to be a lot of rallies so hopefully I'll be ready physically," said Sinner.
Paul, backed by a boisterous home crowd, produced some of his best tennis and defended 10 of the 14 break points he faced, but his hopes of ending a 21-year US drought in the men's Grand Slams were snuffed out by Sinner's precision play.
Paul raised American hopes when he won 11 consecutive points to go up 4-1 in the first set but Sinner retaliated quickly, converting a break point chance in the sixth game and again in the eighth with a wicked forehand winner down the line.
Sinner had a chance to close out the set in the 10th game but could not convert a break point and the crowd broke into chants of "USA!" during a tense tiebreak, where the Italian forced Paul into a backhand error on set point.
Sinner let a break point chance slip through his fingers late in the second set and the umpire had to calm the crowd repeatedly in the tiebreak, as the Italian subdued his opponent from the baseline.
Paul survived a 21-shot rally to save break point in the second game of the third set but a double fault gave Sinner another chance and the world number one converted, whipping a forehand winner past the American.
Sinner pumped his fist with satisfaction after Paul sent a shot long on match point.
"I served very well in the beginning of the match and yes there are some ups and downs obviously," said Sinner.
"Finding my rhythm at the end of the match hopefully helps for the next match."
