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Messi hit yet another landmark with a sublime strike in the first half and his teammate Julian Alvarez mopped up a mistake from Mat Ryan in the second half to put Argentina 2-0 up at Ahmad bin Ali Stadium in Doha.
But Socceroos substitute Craig Goodwin, with a speculative long-range shot which deflected into the net in the 77th minute, revived Australian hopes.
Three minutes later Australia's Aziz Behich went on a daring run, besting four en route to goal, tearing through Argentina's defence his shot blocked to prevent an all-time great World Cup goal.
Garang Kuol then had a shot denied by Argentinian goalkeeper Emiliano Martinez in the seventh minute of added time, the stopper spreading himself wide to prevent the lsat-minute equaliser.
The Australians leave Qatar after their best performance at the World Cup, with two wins in a single edition of the tournament for the first time.
But, just as the only other Socceroos team to advance from the group did in 2006, they depart at the round of 16 phase.
Messi's ninth World Cup goal, and first in the knockout stages of the showpiece tournament, books Argentina a quarter-final against the Netherlands.
The Socceroos were willing but unable to control the brilliant Argentine and his teammates, who played a cautious, controlling gameplan against the Aussies in the first half.
Messi's mob were akin to a prize fighter examining the weakness of their opponent before landing a knockout blow.
It came in the 35th minute when captain Messi started, and finished, an attacking chain.
Darting inside with intent from the right flank, Messi played a sideways pass to teammate Alexis Mac Allister who delivered, one-touch, to Nicolas Otamendi.
In a twinkling of Otamendi's toes, he played back to Messi and with his golden left foot the megastar split three Australian defenders with a shot which gave goalkeeper Mat Ryan, diving to his right, no chance.
The Socceroos, ranked 35 slots lower than the world No.3 Argentines, rarely ventured forward in the first half.
Behich, who irritated Messi with a jumper tug in a first-half flashpoint, charged into the Argentine box after an insightful Jackson Irvine pass in the 19th minute.
But the Socceroos' first forward sortie ended with Behich squeezed over the by-line with the only other glimmer coming from a Harry Souttar header which was cleaned up and away by Argentina.
Argentina doubled their advantage in the 57th minute after an Australian calamity when captain Ryan, receiving a Kye Rowles back-pass, took a heavy touch and two Argentines charged.
Ryan tried to escape the pending danger with his right boot but the ball escaped his control: Alvarez pounced and slotted into the vacant net.
The Socceroos, 20 minutes later, were given a lifeline when Goodwin fired from 20 metres out and found the net.
Substitute Garang Kuol had a late, late chance to level the scores but shot straight at goalkeeper Emiliano Martinez.
