British number two Norrie, 23, beat Hungary’s Fucsovics 7-6 (7-3) 6-3 in one hour and 46 minutes.
He will play Italian world number 96 Lorenzo Sonego in the next round.
Defending champion Rafael Nadal, who had a bye for the first round, made light work of beating fellow Spaniard Roberto Bautista Agut 6-1 6-1.
Earlier, Briton Jamie Murray and Brazilian partner Bruno Soares reached the quarter-finals in Monte Carlo, defeating Edouard Roger-Vasselin of France and Bosnian Ivan Dodig.
The third seeds won 7-6 (7-5) 6-7 (4-7) 10-8 and will play the winners of the tie between John Peers and Henri Kontinen, and Diego Schwartzman and Joao Sousa.