Egypt looked poised to score a famous victory at Atlanta's Mercedes-Benz Stadium after goals from Yasser Ibrahim and Mostafa Zico left them 2-0 up with just 11 minutes of normal time remaining.
But Cristian Romero's 79th-minute header from a pinpoint Messi cross gave the defending champions a lifeline at 2-1.
Four minutes later, Messi volleyed in a superb equaliser - his eighth goal of the tournament - to make it 2-2 and spark wild celebrations among the huge contingent of Argentina fans in the crowd of 68,239.
With extra time looming, Enzo Fernandez nodded in a sensational winner in the second minute of added time to finally break Egypt's resistance.
"Being able to experience moments like this, honestly, I thank God, I'm privileged," Fernandez said afterwards.
"I want to highlight my teammates - we have a phenomenal group, a group that never gives up no matter the difficulties and adversity. We're always together," the Chelsea midfielder added.
Fernandez's goal capped what had been an epic contest, which saw Messi miss a first-half penalty for the second time this tournament, and Egypt have a second-half goal ruled out in a controversial VAR decision.
"We produced a great performance against the world champions," Egypt goalscorer Zico said.
"I don't know what happened in the second half. Strange things happened that everyone saw. It was as clear as the sun in broad daylight."
The win was another nail-biting close shave for Argentina, who had been taken to the wire by Cape Verde in an extra-time thriller in the last 32 on Friday.