The holders looked down and out when goals from Yasser Ibrahim and Mostafa Zico, either side of Messi having a penalty saved, put the Pharaohs in sight of an historic victory in Atlanta.
However, Cristian Romero’s header sparked the comeback as Argentina struck three times in the final 11 minutes plus stoppage time.
Messi smashed home the equaliser to score for a ninth consecutive World Cup game and retake the lead for the Golden Boot with eight this tournament.
Enzo Fernandez completed the turnaround with a superb header two minutes into stoppage time.
At the end, Messi shed tears of joy as he embraced his team-mates.
Egypt were left furious after a number of controversial calls went against them, including a disallowed goal early in the second half, and a member of their coaching staff was red-carded in the aftermath of the winning goal.
But despite a second almighty scare against African opposition, Argentina will face either Colombia or Switzerland in the last eight on Saturday.
Lionel Scaloni attempted to refresh a side that looked dead on their feet in the closing stages of a 3-2 win over Cape Verde in the last 32 on Friday.
Nicolas Tagliafico, Leandro Paredes and Julian Alvarez came in from the team that started in Miami, but Argentina got off to another slow start.
Marwan Attia’s deep cross to the back post was powered home by Yasser.
Argentina had the chance to level five minutes later when Tagliafico was wiped out by Haissem Hassan inside the box.
However, there was another chapter to Messi’s misery from the penalty spot at World Cups.
The eight-time Ballon d’Or’s strike was tentative and Mostafa Shoubir flew to his left to save, meaning Messi has now missed four of his eight non-shootout penalties at World Cups.
He also became the first player to miss two in a single tournament, having also failed to hit the target against Austria in the group stages.
The penalty save was just the start of Shoubir’s inspired afternoon.
Alexis Mac Allster’s powerful header from point-blank range was next to be parried by the Al Ahly stopper.
Shoubir then produced one of the saves of the tournament to prevent Alvarez’s shot finding the bottom corner.
Egypt thought they had doubled their lead on the hour mark after a brilliant counter-attack.
Mohamed Salah played in Mostafa Zico, who dinked the ball perfectly over the onrushing Emi Martinez.
However, Egyptian celebrations were cut short by a hugely controversial VAR intervention for a foul on Lisandro Martinez, just outside the Egypt box, at the start of the build-up to the goal.
The decision finally fired up the majority Argentine crowd, but they were silenced once more moments later by another blistering Egyptian break.
Salah again led the charge before feeding Hassan and his cross was swept in by Zico.
Argentina looked like a beaten side with Messi among those whose heads were bowed during the second-half drinks break.
Yet Romero’s header – that Shoubir just failed to keep out – sparked a remarkable revival as the defending champions hauled themselves off the canvas.
Moments later a dazzling run by Messi teed up Lautaro Martinez, who headed just wide and Messi was to have his moment of redemption seven minutes from time.
Gonzalo Montiel fed the ball back for his captain to strike sweetly on the half-volley.
Shoubir again got a touch but couldn’t get it out as the ball cannoned in off the bar for Messi’s 21st World Cup goal.
Then Fernandez’s bullet header from Lautaro Martinez’s cross completed the fightback and Argentina’s relief was on full display after the final whistle. (AFP)
