The 19-year-old forward's electric pace caused Argentina problems all afternoon and his twin strikes helped France overturn a 2-1 deficit and set up a meeting with Uruguay, who later beat Cristiano Ronaldo's Portugal 2-1 in Sochi.
Messi provided assists for two goals at the Kazan Arena but the 31-year-old was otherwise subdued in the nine”role and his fourth, and possibly final, crack at winning the World Cup was destined to end in disappointment.
While Mbappe was the most influential player on the pitch, Argentina winger Angel Di Maria and France full back Benjamin Pavard fought out a private contest for the best goal of the match with two magnificent long-range strikes.
Di Maria's goal cancelled out Antoine Griezmann's early penalty for France just before the break while Pavard's in the 57th minute levelled the scores at 2-2 after Argentina had edged ahead through Gabriel Mercado.
Mbappe, the first teenager since Brazilian great Pele, in the 1958 final, to score two goals in one World Cup match, then took over to send Argentina slumping out of the finals before the quarter-finals for the first time since 2002.
Twice champions Argentina got a consolation through substitute striker Sergio Aguero in added time but they were simply unable to cope with Mbappe's pace from his first surge towards the box after nine minutes.
Javier Mascherano brought him down and Griezmann crashed the subsequent free kick off the bar but France only had to wait a couple of minutes to open the scoring.
Mbappe set off on another driving run from well inside his own half and Marcos Rojo, well beaten for pace, bundled him to the deck inside the area.
Griezmann's strike from the spot was not as clean as his free kick but goalkeeper Franco Armani went the wrong way and the Frenchman converted his second penalty of tournament.
Argentina looked shell-shocked at Mbappe's early onslaught but Jorge Sampaoli's side was full of big match experience and four minutes before the break they were level.
Di Maria received a pass from the left, took one touch to tee up the ball and hit the sweetest shot into the top-left corner of Hugo Lloris's goal from 40 metres.
Three minutes into the second half and Argentina were ahead, Messi curling a shot hopefully towards Lloris and right back Mercado reacting instinctively to deflect the ball into the net.
The Argentina fans were in full voice but France soon restored parity when Pavard pounced on an over-cooked cross some 20 metres out and lashed it into the net on the half-volley.
Mbappe's first goal came seven minutes later when he snared a loose ball, taking one touch to race past his marker and firing it home with his left foot through Armani's dive.