Medvedev was contesting his second ATP Tour final of 2019, having lost to Kei Nishikori at the Brisbane International last month.
Injury-plagued Jo-Wilfried Tsonga claimed the 17th title of his career on Sunday when he defeated French compatriot Pierre-Hugues Herbert 6-4, 6-2 to claim the Montpellier title.
The 33-year-old Tsonga, who struggled much of last year because of a knee injury, won his last title in October 2017 at the European Open in Antwerp. France's Jo-Wilfried Tsonga in action during the match against Serbia's Novak Djokovic.
Tsonga next plays Radu Albot of Macedonia, who won 6-2, 7-6 (2) against qualifier Marcos Baghdatis of Cyprus.
Medvedev put Fucsovics under pressure early on and had squandered three break points by the time he made one stick to go 4-3 in front. At the Cordoba Open, Guido Pella set up an all-Argentine final after beating Uruguayan Pablo Cuevas.
Pella took the first set in just 27 minutes, but he was broken in his first service game of the second set.
Russian third seed Daniil Medvedev dispatched French seventh seed Gael Monfils 6-2, 6-4 to claim a spot in the final in Bulgaria.