British number two Edmund, 22, beat world number 122 Fucsovics 6-2 6-1 in North Carolina.
Edmund, 45th in the rankings, will play world number 46 Steve Johnson in the next round.
Meanwhile, Britain's Laura Robson lost 6-2 6-4 to Russian Alla Kudryavtseva in the first round of US Open qualifying.
There was better news for fellow Britons Naomi Broady and Katie Boulter who are through to the second round and are two wins away from the main draw.
Broady beat Switzerland's Jil Teichmann 6-2 4-6 6-1 while Boulter beat American Danielle Collins 6-4 6-0.
Britain's Kyle Edmund is through to the quarter-finals of the Winston-Salem Open after a straight sets win over Hungary's Marton Fucsovics.