Russian Medvedev suffered a jarring early exit from the Canadian Masters this month but overcame early mistakes at the tune-up tournament for the year's final major, trading breaks with Shapovalov before breaking him again in the 11th game.
A frustrated Shapovalov received a warning for whacking his racket as Medvedev, who won the tournament in 2019, wrapped up the first set.
Medvedev dropped just four first-serve points in the second set and never faced a break as Shapovalov saved five break points before losing his serve in the penultimate game.
"The match was pretty tight, I felt like it was (a) great level," Medvedev said. "My goal was to stay there every point possible."
The Russian top seed was joined in the last eight by Stefanos Tsitsipas, Felix Auger-Aliassime and Borna Coric. He faces Taylor Fritz (24) in the next round, after the Indian Wells winner came back from a set down to beat Russian Andrey Rublev (24) 6-7(4) 6-2 7-5 on Centre Court.
Rublev's game deteriorated just as Fritz found his form, the American breaking in the third and fifth games of the second set and hitting a dozen winners, twice as many as his opponent.
Fritz saved three break points in a tense sixth game of the decider before converting his own break point in the 11th.
"Obviously, I want to go even further, semis, finals, win the title but to put another Masters quarter-final on the record for the year helps a lot," said the Indian Wells champion, who stands on the brink of entering the top 10 for the first time.
Third seed Carlos Alcaraz, a quarter-finalist at Flushing Meadows last year, beat former U.S. Open champion Marin Cilic 7-6(4) 6-1 to set up a meeting with British world number 11 Cameron Norrie, who downed American Ben Shelton 6-0 6-2.
Croatian Coric backed up his stunning victory over Rafael Nadal late on Wednesday by taking out another Spaniard in Roberto Bautista Agut.
Former world number 12 Coric is still on the comeback trail after shoulder surgery but showed he should not be underestimated at the US Open with a clinical 6-2 6-3 victory.
A quarter-finalist at Flushing Meadows in 2020, Coric will next face Canadian seventh seed Auger-Aliassime, who saved two match points in the second set in a stunning 2-6 7-6(1) 6-1 comeback victory over Italian Jannick Sinner.
Greek Stefanos Tsitsipas (24) had few problems taking down Diego Schwartzman (30) 6-3 6-3 in a little over 90 minutes after the Argentine served up nine double faults.
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