The British No1 was playing his first match since his breakthrough triumph at Indian Wells 10 days ago and came out on top against the Hungarian for his seventh victory in a row.
It was also his 11th from the last 12 matches as he continues to enjoy the form of his life, overcoming a difficult start to win 7-6 (4) 6-1 in Vienna.
The world No14 will play sixth seed Felix Auger-Aliassime in the next round.
Norrie’s landmark win in the Californian desert had significantly bolstered his standing in the game, but it looked like he might be brought back down to earth as Fucsovics started well.
Along with some power hitting, Norrie’s radar was off and a string of errors allowed Fucsovics to open up a 3-0 lead.
Norrie began to find his level, though, and from 4-2 down reeled off four successive games to allow him to serve for the first set.
Fucsovics broke back to love and the opener had to be settled by a tie-break, which Norrie got the better of, sealing it after his opponent dragged a forehand wide.
Fucsovics’ spirit was broken at that point and Norrie doubled down on his superiority, racing into a 5-0 lead in the second set.
Fucsovics finally got on the board but Norrie saw it out to join fellow Briton Andy Murray in the last 16.
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