In the first match, France roared back to beat Argentina 4-3, with Mbappe scoring twice.
Later in Sochi, Ronaldo — who started the World Cup with a hat-trick against Spain — was powerless to prevent Portugal slipping out of the tournament as Edinson Cavani scored twice in a 2-1 win to send the European champions home.
Ronaldo, who will be 37 by the time of the next World Cup, was unable to keep Portugal alive despite incessant pressure from his side in the dying stages of the match.
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.
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.