The Russian navigated an awkward first set before eventually triumphing 7-6, 6-3 and will face either compatriot Andrey Rublev or Ugo Humbert of France in the next round.
Medvedev led 5-2 in the first set but errors crept in and he needed a tie-break to see off a stirring comeback from the 12th-ranked De Minaur.
He found his range in the second set, securing an early break before taking his second match point with a fierce volley.
Earlier, Nicolas Jarry battled to a marathon victory to set up a potential quarter-final with Alexander Zverev, who plays later on Saturday.
The unseeded Chilean triumphed 6-7 (4/7), 7-6 (7/4), 6-3 over Italy's Matteo Arnaldi in a three-hour match.
Tournament favourite and world number two Carlos Alcaraz will play Italy's Lorenzo Musetti in round two after dispatching qualifier Yannick Hanfmann on Friday.
The China Open is taking place for the first time since 2019 after the government lifted its strict zero-Covid policy.
Most of the top male players are in action with the notable exception of Novak Djokovic.