Italy’s Sinner beat home hope Paul 7-6 (7-3) 7-6 (7-5) 6-1 in front of a rowdy New York crowd in the night session on Arthur Ashe.
Russia’s Medvedev, the 2021 US Open champion, sent an emphatic statement earlier on Monday with a 6-0 6-1 6-3 win over Nuno Borges of Portugal.
The pair are the only Grand Slam champions remaining in an open men’s singles draw.
Sinner came back from two sets down against Medvedev to clinch the Australian Open title in January.
He has won five of his past six meetings against Medvedev, having lost six in a row previously, but Medvedev won their five-set Wimbledon quarter-final in July.
Sinner grew into the match against Paul, with his forehand initially leaking errors before the 23-year-old started to outmuscle his opponent from the baseline.
He trailed 14th seed Paul by a double break in the first set before reeling off three games in a row and dominating the resulting tie-break.
The second set was similarly close, but an early break in the third set Sinner up to close out victory in two hours and 42 minutes.
The only slight concern for Sinner was when he stumbled in a rally and clutched his hip – a problem he has struggled with throughout the year.
“There’s going to be a lot of rallies so hopefully I’ll be ready physically,” Sinner said of his match against Medvedev.
“We’ll try to do our best on court and hopefully you guys enjoy it.”
Sinner has lost just five matches this season, with two of those defeats coming on a hard court.
His most recent success was at Cincinnati in August, winning the title the day before news emerged he had been cleared of any wrongdoing after twice testing positive for a banned substance earlier this year.