Medvedev earned a straight-sets win over the 2019 winner, but the French lucky loser battled hard before the Russian claimed a 7-6(6), 6-4 victory to kick-start his grass season.
Mannarino broke serve in the opening game with Medvedev responding immediately, and the Russian came out on top in the tiebreak after saving two set points. Medvedev made the decisive break in the second set to lead 5-4 and held serve for the win.
The pair came into the meeting level at 4-4 in their head-to-head, but Medvedev has now won three on the trot against Mannarino and will next meet lucky loser Reilly Opella after the American beat Chile's Nicolas Jarry 7-6(5), 6-3.
French second seed Ugo Humbert beat Briton Daniel Evans 7-5, 6-3 and in the last eight will face Portuguese seventh seed Nuno Borges, who got a walkover after Finn Otto Virtanen failed to recover from a knee injury picked up during his first-round win.
Poland's Hubert Hurkacz also withdrew through injury, sending Estonian qualifier Mark Lajal into the quarter-finals, where he will meet Zizou Bergs after the Belgian beat Australian fourth seed Alexei Popyrin 6-7(3), 6-2, 7-6(3).
See all the men's results and the draw here.
Mertens and Andreescu advance
Belgium's Mertens had little trouble seeing off Greece's Sakkari 6-3, 6-0, and her win versus the former world number three earned her a quarter-final meeting with China's Yuan Yue after the lucky loser's 6-4, 6-3 win over Australian Kimberly Birrell.
Wildcard Canadian Andreescu eased to a 6-0, 6-4 victory against New Zealand's seventh seed Lulu Sun, and last year's losing finalist will meet qualifier Elena-Gabriela Ruse of Romania in the last eight.
Ruse had a 6-4, 6-3 win over fellow qualifier Carson Branstine of Canada, who had knocked out defending champion Liudmila Samsonova in the first round.
Suzan Lamens of the Netherlands beat American Ann Li 7-6(4), 7-6(5) and will next face Italy's Elisabetta Cocciaretto.