Ben Yedder shrugged off two defenders to rifle home in the 22nd minute and the striker had two further efforts ruled out for offside.
Kalifa Coulibaly missed a pair of easy chances for the home side but Ben Yedder was rewarded for his persistence when he raced onto a ball from Aleksandr Golovin and hit a rising shot past Alban Lafont.
Moses Simon missed a good chance to bring Nantes back level before Ben Yedder had a second effort ruled out for offside just before the half-hour mark.
Nantes pressed after the break with Kalifa Coulibaly hitting the post and Monaco goalkeeper Bernard Lecomte saving well to deny Kader Bamba.
Ben Yedder thought he had scored his second on the hour mark but after tapping home from a cut-back from Islam Slimani he saw another effort narrowly ruled out.