Medjedovic, ranked 88th in the world, broke for a 5-3 lead in the first set which he served out to love.
De Minaur broke and consolidated for a 3-0 advantage in the second set but the 22-year-old qualifier battled his way back, winning the last five games to secure two breaks and triumph.
The Australian committed five double faults on a difficult afternoon and converted only one of nine break points.
The 2023 Next Gen Finals champion will next face Nuno Borges after the Portuguese defeated Tomas Martin Etcheverry 6-3, 7-6 (7/4) to progress to the last eight.
Later Wednesday, seventh seed Cameron Norrie faces Ethan Quinn and and rising Spanish star Rafael Jodar takes on Camilo Ugo Carabelli.
On Thursday, world number two Carlos Alcaraz is scheduled to take on Tomas Machac in another round of 16 clash, but the Spaniard skipped training on Wednesday.
Alcaraz, who lost the Monte Carlo Masters final to new world number one Jannik Sinner on Sunday, defeated Otto Virtanen on Tuesday in the first round.
During that match, the top seed was treated for a problem with his wrist and forearm, which he said he hoped would not become a problem.
