In his 1,000th career match, Messi scored the opening goal with a sublime effort before Julian Alvarez scored in the second half to set up a quarterfinal tie against the Netherlands and knock Australia out of the tournament, as the Socceroos were unable to build upon their fortuitous late goal.
"I am very happy for taking another step forward, achieving another objective," Messi said. "It was a very strong and difficult match -- we knew it was going to be this way.
"We didn't have much time to rest up and we were concerned as we knew it was going to be a physical match and they were very strong. Luckily we found the goal and we were one goal ahead, and then two goals ahead.
"And then we had a setback with the goal -- but that's the World Cup for you, the matches are difficult and we got the win."
Despite Argentina's firepower, Australia managed to limit the early proceedings, keeping them uneventful with neither side exerting any dominance.
But the Socceroos' defence was breached on 35 minutes as Messi showed his undeniable quality to score his first-ever goal in the World Cup knockout stage.
The Paris Saint-Germain forward started the move with a darting run inside from the right wing -- a quick exchange of passes landed him inside the box before he unleashed a left-footed curling shot past Mat Ryan.
"He [Messi] is incredible. One of the greatest ever," Australia boss Graham Arnold said. "We really worked hard not to be in awe of him, because of the great player he is.
"But wow, he's remarkable. I had the privilege of playing against Diego Maradona, and now coaching against Messi -- they're both wonderful players and Argentina should be so proud and happy to have players of that calibre."
Buoyed by rapturous support, Argentina tightened their grip on a quarterfinal spot 12 minutes after half-time as Rodrigo De Paul pounced on a slack touch by Australia goalkeeper Ryan, allowing Alvarez to slot into a vacated net, doubling his side's lead.
A triple substitution for Australia on 72 minutes signified a desperate last roll of the dice by manager Arnold and they were handed a lifeline when an own goal by Enzo Fernandez halved their deficit 13 minutes from time.
Craig Goodwin latched onto a loose ball outside the box and struck a fierce effort first-time, which took a huge deflection off Fernandez, fooling goalkeeper Emiliano Martinez and hitting the back of the net.
With renewed optimism, Aziz Behich came close to levelling the game three minutes later as he embarked on a fantastic darting run, beating four Argentina defenders before eventually being thwarted by Lisandro Martinez in the box.
Despite their desire, a comeback didn't materialise for Australia and manager Arnold was full of praise for their impressive run in the tournament.
"They're extremely disappointed -- it's not every day you get to play the third-best nation in the word and play those types of players," he said.
"I felt we did a good job. I wanted to say how proud I am of them and the sacrifices they've made through the campaign. Everyone said before we came here we were the worst Socceroo team ever and to qualify for a World Cup -- but that's gone now -- we've done exceptionally well."