The big spenders, who sold World Cup top scorer James Rodriguez to Real Madrid in July, had lost their opening two Ligue 1 encounters and rode their luck at times in Nantes.
Monaco goalkeeper Danijel Subasic gave away a penalty midway through the first half but saved Serge Gakpe's spot kick.
Falcao, who missed out on helping Rodriguez lead Colombia to the World Cup quarter-finals because of injury, is still feeling his way back to full fitness but planted an unstoppable header into the net on the stroke of halftime.
The goal followed great work by full back Layvin Kurzawa, who was excellent all match for Leonardo Jardim's side.
Monaco are 17th in the table, which is led by Bordeaux after Saturday's 3-1 victory at Nice made it three wins out of three. Stuttering champions and title favourites Paris St Germain are sixth on five points following Friday's 0-0 draw at Evian.
(Writing by Mark Meadows, editing by Pritha Sarkar)
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