Messi, 35, was winning his 169th cap for his country and they needed his brilliance to take them through to the next round and a tantalising tie against the Netherlands on Friday.
Argentina had barely threatened in the first half but Paris St-Germain's Messi stroked in a delightful low finish to spark wild celebrations from their fans.
It was his 789th career goal, 94 of those coming for Argentina, and Messi is now the joint top goalscorer in Qatar.
The South American supporters had been on their feet singing all game and they had further delight when Julian Alvarez punished Mat Ryan's mistake to double their advantage.
Australia had offered very little but surprisingly pulled one back with 13 minutes remaining when substitute Craig Goodwin's strike took a huge deflection off Enzo Fernandez.
They could have levelled shortly after through Aziz Behich, whose incredible solo run took him past four players, but his shot was superbly blocked by Lisandro Martinez's sliding challenge as Argentina edged through.
Australia frustrated Argentina for 35 minutes, sitting deep and compact, allowing no space in behind - Alejandro Gomez's wild shot that flew way over was the only sniff they had.
But every time Messi got his left boot on the ball there was a buzz of excitement in the stadium, sharp intakes of breath anticipating something special.
And the little magician showcased exactly why he is considered one of the greatest of all time in his landmark game, starting and finishing a move that had blue and white shirts dancing in the stands.
Messi saw a free-kick from the left cleared but Argentina recycled the ball, Brighton's Alexis Mac Allister fizzing a pass into Nicolas Otamendi, who laid it off for Messi to take a touch to set himself before caressing a precise finish into the bottom corner.
Arms outstretched, Messi ran away to celebrate in front of his adoring faithful after converting his ninth World Cup goal - remarkably his first in a knockout-stage match - to move just one behind Gabriel Batistuta's Argentina record.
A trademark jinking run then came to nothing, but thrilled those watching, who broke out with chants of "Messi, Messi, Messi".
He could have drawn level - or even surpassed Batistuta - in the second half, but slipped at the vital moment to scuff a shot straight at Ryan and fired another two efforts off target.
Ryan, the former Brighton keeper, had a moment to forget on his 10th World Cup appearance - the joint most for his country - when he took a heavy touch from a back-pass, allowing Manchester City's Alvarez to nip in, spin and score into an open net.
Following their group-stage heroics, Australia were looking to spring another surprise and had hope courtesy of Goodwin's 20-yard drive that took a fortuitous deflection with 13 minutes remaining.
They could have taken the game to extra time when full-back Behich embarked on a mazy run to beat numerous defenders, but Martinez reacted in the nick of time.
An even better chance fell to the Socceroos in the 97th minute when Garang Kuol brought the ball down in the area but his shot on the turn was well kept out by Emiliano Martinez's reflex save.
-- Courtesy of BBC Sport