France, the 1998 champions, looked a totally different side from the one that struggled to find their cutting edge in the group stage, pouring forward with pace and purpose to stretch the ageing Argentine defence in Kazan.
Antoine Griezmann gave France the lead from the penalty spot but the South Americans levelled with a sweet hit from Angel Di Maria and edged ahead shortly after half-time through Gabriel Mercado.
But defender Benjamin Pavard equalised with a thunderous strike to pull France level and once more change the complexion of the game.
That set the scene for 19-year-old Mbappe, who netted two goals in four second-half minutes to become the first teenager to score at least twice in a World Cup match since a 17-year-old Pele in 1958.
Sergio Aguero gave Argentina late hope but they ran out of time and head home after a rollercoaster ride in Russia that ultimately ends in bitter disappointment for the two-time former champions.
Man-of-the-match Mbappe brushed off comparisons with Brazil great Pele.
“I’m very happy, and it’s flattering to be compared to a great player like Pele,” he said.
“But he’s in another category. Still, it’s great to join the list of players that have achieved such feats.”
Mbappe was born in 1998, the year Les Bleus secured their first World Cup title under captain Didier Deschamps.
Deschamps, now national manager, praised his team’s character to fight back from 2-1 down.
“If I am a coach it is because I want to live this: a full stadium, highly experienced teams. Our team is much younger, but it’s there,” Deschamps said.
“We answered the call, we have a lot of character and it was not easy as we were leading, then there was an equaliser, then we were 2-1 behind, but we kept fighting.
“There is an excellent mentality in this group and we did everything to go further. Since I am responsible for everything, particularly when it doesn’t go well I’m very proud.”
Beaten finalists four years ago, Argentina only reached the last 16 by the skin of their teeth after a shambolic group phase and despite a brave effort against France, the age of their players and an unbalanced squad ultimately caught up with them.
Messi has spent the tournament trying to hold his squad together, even apparently bypassing coach Jorge Sampaoli to give coaching instructions in the 2-1 win against Nigeria that saved the South Americans from a humiliating group-stage exit.
He has scored only once and Argentina’s campaign so far was defined as much by Diego Maradona’s erratic and emotional outbursts.
The Barcelona star, at the age of 31, may have played his last World Cup game in a career curiously unfulfilled at international level despite his astonishing achievements with the Spanish giants and his multiple individual awards.