Goffin, the world number seven, won in three sets, 7-5, 6-3, 6-1 in front of a huge crowd of close to 26,000 at the Stade Pierre-Mauroy in the northern French city of Lille.
French number one Jo-Wilfried Tsonga faces Steve Darcis in the second singles rubber later as the hosts look to stay in the tie and win the trophy for the first time since 2001.
Goffin, who last week lost to Grigor Dimitrov in the title decider at the ATP Finals in London, needed just under two hours to give the Belgians the upper hand as they look to win the Davis Cup for the first time.
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