The result left France top with 12 points, six ahead of second-placed Greece, who had Konstantinos Mavropanos sent off in the 69th minute after Mbappe's 55th minute spot kick.
"We wanted to follow up on our win against Gibraltar, in front of our crowd," said midfielder Eduardo Camavinga, referring to France's 3-0 win on Friday.
It came after earlier opportunities for Mbappe and Kingsley Coman but the offensive trio hit a brick wall.
Vlachodimos parried Mbappe's spot kick but it had to be taken again as the keeper had left his goal line too early.
After Saturday's win over Gibraltar, France have now scored nine goals and kept clean sheets in all four qualifying games.
Persons:
Kylian Mbappe, Mbappe, Just Fontaine, Erling Haaland, Konstantinos Mavropanos, Didier Deschamps, Eduardo Camavinga, Randal Kolo Muani, Kingsley Coman, Les Bleus, Jules Kounde's, Odysseas Vlachodimos, Antoine Griezmann, Greece's, Mavropanos, Kolo Muani, Saturday's, Julien Pretot, Clare Fallon, Ken Ferris
Organizations:
Greece, St Germain, Manchester City, Gibraltar, Mbappe, Thomson
Locations:
France, Norway, Cyprus, Greece, Gibraltar, Ireland, Netherlands