The PSG forward is the first teenager to score twice at a World Cup since Pele in 1958, with both goals coming within four second-half minutes to cancel out Angel Di Maria and Gabriel Mercado's goals.
Antoine Griezmann had put France ahead from the penalty spot after just 12 minutes, Mbappe's impressive run being ended by Marcos Rojo inside the area.
Argentina were struggling to make any sort of impact for the majority of the first half until Di Maria beat Hugo Lloris from more than 30 yards just before the break.
The momentum remained with the South Americans after the break and Mercado directed Lionel Messi's shot past helpless Lloris minutes into the second half.
Their lead did not last long as Benjamin Pavard's half-volley bent around Franco Armani for the equaliser. Then Mbappe stepped up.
First making the most of hesitation in the penalty area to beat Armani from close range before running on to Olivier Giroud's through ball to find the far corner.
Sergio Aguero, on as a substitute, headed home in the 93rd minute but Argentina's fightback came too late.