Medvedev did not face a single break point in defeating the 28-year-old Greek 6-4, 7-5.
"I felt like I was playing very well today," said Medvedev, the fifth seed.
"I put a lot of pressure on him in the first set.
"Second set different story. Very tight one. I'm happy that I managed... to get some good returns, get some good rhythm," the 30-year-old said.
The Russian broke his opponent's serve to go up 6-5 in the second set and closed things out in one hour and 24 minutes.
The players were meeting in the singles draw for the 16th time on Tour, with 6th-ranked Medvedev improving his head-to-head record to 11 wins.
Medvedev won the Shanghai tournament in 2019.
His next opponent, Paul, meanwhile, beat Zizou Bergs 7-6(5), 6-4.
"I know sometimes it's hard to back up a big win," Paul said of the 39th-ranked Belgian, who upset second-seeded Carlos Alcaraz in the previous round.
Though Paul said Bergs "raised his level" in the second set, narrowing a 0-4 deficit to trail by just one game, it was the American who triumphed in the end.
Jodar sets up Ruud meeting
Rising Spanish star Rafael Jodar earlier beat Botic van de Zandschulp 6-7(4), 6-4, 6-2 to secure his quarter-final spot.
It was not without incident, though, with Jodar handed a code violation in the second set for swearing at a ball boy in Spanish when asking for a towel.
Jodar will play Casper Ruud in the quarter-finals. The Norwegian advanced when fourth-seeded Felix Auger-Aliassime retired injured when down 2-5 in the first set.
Ruud, ranked 24th in the world, said his opponent's trouble appeared to start in the fourth game of the first set.
"It seemed to me that he maybe twisted his knee... it looked like he kind of stumbled a bit with a wrong step," Ruud told reporters.
"From that moment it seemed like he was in a lot of pain," he said.
Auger-Aliassime needed a medical timeout after the fifth game, starting the next game down 1-4 with white strapping around his left knee.