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 "false 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.
"If I am a coach it is because I want to live this - a full stadium, highly-experienced teams," France coach Didier Deschamps said. "Our team is much younger, but it's there. We answered the call, we have a lot of character and it was not easy ... but we kept fighting."
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. His second came after 68 minutes at the end of a flowing French move with Olivier Giroud providing the subtlest of touches to lay the ball into Mbappe's path and the teenager finishing right-footed.
"It's flattering to be the second one after Pele, but let's keep things in context," Mbappe said. "Pele is another category, but it's good to be among these people." Argentina continued to fight over the last 20 minutes but Aguero's header from Messi's cross in stoppage time was too little, too late. "This is very painful, especially because the players have made a huge effort," said Sampaoli, already under pressure after Argentina's stuttering start to the tournament. "It was a very difficult game but we have not achieved our goal here in Russia. I'm sad, I'm frustrated - I think that's quite normal. I won't decide my future today."
Portugal exits the World Cup
Uruguay forward Edinson Cavani scored a stunning brace to lead the South Americans to a 2-1 win over European champions Portugal in the World Cup round of 16 on Saturday and set up a quarter-final against France.
Cavani switched play with a long cross to Luis Suarez and continued his run to head home a pinpoint deep cross from the Barcelona forward and give Uruguay a seventh-minute lead. Portugal defender Pepe made Uruguay pay for sitting back when he leapt unmarked to head in a cross from Raphael Guerreiro following a short corner to level the match at 1-1 in the 55th minute.
That was the first goal the twice World Cup champions had conceded in the tournament in Russia, but it took Cavani only seven minutes to put them back ahead at the packed Fisht Stadium overlooking the Black Sea. The Paris St Germain forward curled in a first-time right-foot shot from Rodrigo Bentancur's pass to beat Portugal goalkeeper Rui Patricio before going off injured in the 74th minute as the Uruguay defence dug in to hold on to their lead. "Happy for what happened today, I hope we can go further," Cavani said.
Portugal, who won Euro 2016 in France, dominated possession but their attack did not have enough bite to break through the Uruguay defence twice, leaving captain Cristiano Ronaldo a frustrated figure during the entire match.
The Real Madrid forward could not add to his four goals in the first two matches, failing to improve his record of failing to score in all six World Cup knockout matches he has played. Ronaldo had an opportunity in the first half to score from a free kick about 30 metres out but could only hit the wall. "It's not a fair result, we did enough for another result," Portugal coach Fernando Santos said. "Uruguay scored with their first shot on our goal. We are very sad but that's football." Uruguay will play France in the last eight in Nizhny Novgorod on Friday. –Reuters
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