A stunning burst of pace from the youngster earned a penalty for his side in the early stages before a quick-fire double after the break put his team in control.
France now advance through to the quarter-finals where they will face either Portugal or Uruguay, who play later on Saturday evening.
Both teams started brightly with tackles flying in from all angles and the first shot almost led to a goal, with Griezmann striking the angle between bar and post from 25 yards from a free kick.
Two minutes later, France were awarded a penalty after a sensational burst of pace from Mbappe left Javier Mascherano in the centre of the pitch before Marcos Rojo opted to haul down the Paris Saint-Germain attacker inside the box.
The foul gave Griezmann the opportunity to score from 12 yards and the forward made no mistake as he slotted the ball into the centre of the goal as Franco Armani dived to his left.
Argentina were having their moments at the other end of the pitch but the defence were anxious whenever Mbappe got hold of the ball in space and it resulted in Jorge Sampaoli's team having to sit deeper than perhaps planned.
France appeared to be comfortable heading into half time but just before the break, Di Maria got Argentina back on level terms in some style with the 30-year-old drilling a long-range strike into the top corner of Hugo Lloris's net.
Soon after the restart, Argentina found themselves ahead but unlike Di Maria's opener, there was an element of luck to it as Lionel Messi's shot from the edge of the area took a deflection off Mercado before finding the bottom corner.
Going ahead resulted in Argentina over-committing in tackles and with the pace of their play and it only led to their energy diminishing quickly, which played into France's hands.
However, the equaliser for Les Bleus would have taken the breath away of any side. A cross from the left went through to Pavard who was waiting on the edge of the penalty area and the full-back struck an exquisite first-time half-volley into the far corner from 20 yards out.
While that strike was a goal of the tournament contender, Mbappe's quick-fire double was ruthless and one of the world's biggest prospects deserved to take centre stage.
Mbappe's first came as a result of a half-clearance, with the 19-year-old's first touch taking him into a shooting position before he drilled a low shot underneath Armani from 10 yards.
Four minutes later, Mbappe effectively ended the contest. A fine team move concluded with Giroud passing the ball through to the advancing winger, whose first-time effort flew past Armani and into the far corner of the net.
With a quarter of the match remaining, Argentina looked done and even the introduction of Aguero failed to produce an immediate response, with France looking more likely to net a fifth.
As the match progressed towards a close, Messi nearly produced the bit of individual brilliance which had been lacking but after dancing past two France defenders, the Barcelona maestro fired straight at Lloris from 14 yards.
In the third minute of added-on time, Argentina did halve the deficit through Aguero's header at the back post but it proved too little too late as the South Americans bowed out at the first knockout stage.
FRANCE (4-2-3-1): Lloris; Pavard, Varane, Umtiti, Hernandez; Kante, Pogba; Mbappe (Thauvin 89'), Griezmann (Fekir 83'), Matuidi (Tolisso 75'); Giroud
ARGENTINA (4-3-3): Armani; Mercado, Otamendi, Rojo (Fazio 46'), Tagliafico; Mascherano, Perez (Aguero 66'), Banega; Di Maria, Pavon (Meza 75'), Messi