France came close to taking the lead inside the first minute when Mbappe danced his way into the Netherlands box before firing towards goal, but Jasper Cillessen was on hand to make the save.
Quincy Promes had an opportunity down the other end moments later, but the attacker's low strike just missed the post during a very open start to the Nations League clash.
Mbappe and Cillessen again went head to head in the 11th minute when the former met a clever pass from Antoine Griezmann, and the Barcelona stopper made another save.
France did take the lead in the 14th minute, however, when Mbappe tapped a low Blaise Matuidi cross into the back of the net. The goal came from a huge mistake from the visitors though, with Promes's loose header falling straight into the path of Matuidi, and Mbappe simply could not miss from close range.
France had the chance to double their lead in the 18th minute when Griezmann found Atletico Madrid teammate Lucas Hernandez with a stunning through-ball, but the left-back lifted his effort high and wide.
Mbappe then flicked a header into the arms of Cillessen in the 26th minute as France continued to create chances.
France's next opportunity fell for Samuel Umtiti 10 minutes before the break, but the centre-back's header was straight at Barcelona teammate Cillessen, and Mbappe's close-range finish was all that separated the two teams at the interval.
Mbappe had the first serious attempt of the second period in the 51st minute, but the forward's strike from distance was always high of the Netherlands goal. Griezmann then struck a low attempt wide of the post just before the hour during a low-key period of the match.
Netherlands had a golden chance to level the scores in the 66th minute when Wijnaldum broke into France box, but the Liverpool midfielder rolled the ball wide of the far post.
Netherlands were the team on the front foot in the period that followed, and Memphis Depay struck a powerful effort into the side-netting with the visitors threatening a second.
France had a chance of their own in the 73rd minute when Mbappe delivered low towards Giroud following a swift break, but Virgil van Dijk was on hand to make the clearance.
Giroud did fire France ahead in the 75th minute, however, with the striker's excellent volley from a Benjamin Mendy cross finding a route into the back of the net.
FRANCE (4-2-3-1): Areola; Pavard, Varane, Umtiti, Hernandez (Mendy 62'); Pogba, Kante; Mbappe, Griezmann (N'Zonzi 81'), Matuidi; Giroud (Dembele 89')
NETHERLANDS (3-4-3): Cillessen; Van Dijk, Blind, De Ligt; Tete (Janmaat 82'), Propper, F De Jong, Babel (L De Jong 88'); Promes, Memphis, Wijnaldum (Vormer 76')