Teenager Mbappe was in devastating form in the heavyweight clash in Kazan, scoring two second-half goals to bring Argentina's tumultuous campaign in Russia to an end.
The Paris Saint-Germain forward's surging 70-yard burst also led to the opener after 13 minutes, Antoine Griezmann slotting home from the penalty spot after Mbappe had been hauled down by Marcos Rojo.
Yet Argentina turned the game around to go ahead, Angel Di Maria's stunning long-range strike before the break followed by Gabriel Mercado's fortuitous effort early in the second half.
However, Benjamin Pavard matched Di Maria's effort with an outstanding equaliser in the 57th minute before Mbappe took centre stage, securing Didier Deschamps' side a place in the quarter-finals, where they will play either Uruguay or Portugal.
Messi – who will face questions over his international future in the aftermath of the defeat - did set up a late header for Sergio Aguero but it was not enough to save Argentina, with the beaten finalists from 2014 slipping out of this year’s tournament.
France, who made six changes from the side that drew 0-0 with Denmark last time out, were almost ahead after nine minutes, but Griezmann's wonderful 25-yard free-kick crashed back off Franco Armani's crossbar.
The Atletico Madrid striker did not have to wait long to open the scoring, though - largely thanks to the scintillating pace of Mbappe.
After picking the ball up deep inside his own half, he set off on a sparkling run that was only halted when Rojo clumsily hauled him down just inside the area.
Griezmann stepped up and coolly converted from the spot, sending the ball low to Armani's right for his second goal - both penalties - at the World Cup.
Argentina had scarcely offered anything in the way of an attacking threat during a tepid first half, yet drew level four minutes before the interval when Di Maria unleashed from distance with his left foot to beat Hugo Lloris.
Boosted by the equaliser, Jorge Sampaoli's team started the second period brightly and were duly rewarded with a second goal three minutes after the restart.
Ever Banega's free-kick was headed out towards Messi and the Barcelona star's whipped effort towards goal was diverted past a helpless Lloris by Mercado, who had stuck out his foot in an apparent attempt to control the ball.
Their lead lasted just nine minutes, though, as France pulled level with a goal of the highest quality.
Lucas Hernandez's cross from the left found his full-back partner Pavard, who sliced across his half-volley from outside the area, sending it spinning into Armani's top-right corner.
Mbappe put France ahead again in the 64th minute when he picked up a loose ball in the area, found a yard of space and fired in low past Armani, who was slow getting down. His second - a first-time finish - followed just four minutes later, finally putting clear daylight between the teams.
Aguero set up a tense finale by nodding home in the third minute of stoppage time, but it was not enough to prevent Argentina - and Messi - from crashing out.
Key Opta stats:
- Lionel Messi has played 756 minutes of football in World Cup knockout stage matches, but has failed to score a single goal.
- Les Bleus are unbeaten in their last nine World Cup meetings with South American nations (W5 D4), since losing 1-2 against Argentina in the 1978 tournament.
- This was just the fifth time Argentina have conceded 4+ goals in a World Cup match, and first since losing 0-4 to Germany in the 2010 quarter-final.
- Argentina are the first team to score at least three goals but still lose a World Cup match since the USSR vs Belgium in 1986
- France’s Kylian Mbappe became the first teenager to score at least twice in a World Cup match since Pele netted twice for Brazil against Sweden in the 1958 final.
- France have never lost an international match in which Antoine Griezmann has scored for them (W17 D2 L0).