Lionel Messi's first goal in the knockout rounds of the World Cup sent Argentina ahead in the 35th minute, before Julian Alvarez doubled the advantage for the South Americans just before the hour.
Australia benefitted from a huge deflection in the 77th minute to halve the deficit, with Craig Goodwin's strike hitting Enzo Fernandez before dropping into the back of the net, but the Socceroos could not find a second.
Lionel Scaloni's side will take on the Netherlands in the final eight on December 9, with Holland beating the USA earlier in the day to secure their own position in the quarter-finals.
Australia finished second in Group D behind France to advance to the round of 16 for just the second time in their history, but it proved to be a match too far for Graham Arnold's side.
Argentina had a shout for a penalty in the fourth minute when Papu Gomez's shot hit the arm of Keanu Baccus, but the play continued without any punishment for the Australians.
Gomez then had Argentina's first serious attempt in the 17th minute, but his effort from a wide angle was always rising over Mathew Ryan's goal, with Australia relatively comfortable in the early stages.
The Socceroos were causing problems with their physicality from set pieces, with Harry Souttar a particular threat inside the penalty box.
For long spells of the first half, it was a struggle for Argentina, but the South Americans took the lead in the 35th minute of the contest, with Messi bursting into life.
There did not appear to be much danger when the 35-year-old picked up the ball inside the Australian box, as there were plenty of defenders between him and the goal, but the forward managed to work some space before feeding the ball into the bottom corner of the net for 1-0.
Messi's goal was the standout moment in an otherwise uneventful first period, with Australia finding it difficult to carve out chances against Scaloni's side.
Ryan had to keep out a strike from Messi in the 50th minute, before Argentina had a nervy moment down the other end, with Mitchell Duke almost beating Emiliano Martinez to a short back-pass.
However, Argentina doubled their advantage in the 57th minute through Alvarez, who took advantage of a mistake from Australia goalkeeper Ryan to find the back of the net.
Ryan attempted to drive past two Argentine players inside his own box and instead lost the ball to Alvarez, who simply could not pass up the chance to send his side into a two-goal lead.
Nicolas Tagliafico fired wide of the post 15 minutes from time after both managers made changes, but it was Australia that scored in the 77th minute when Goodwin's shot took a huge deflection off Fernandez before nestling in the back of the net.
Behich had a brilliant chance to level in the 80th minute when he broke into the Argentina box, but Lisandro Martinez was across to make an excellent last-ditch challenge.
Lautaro Martinez should have put the game behind Australia in the 89th minute, but he somehow lifted the ball over the crossbar from a narrow angle, before the Inter Milan forward was denied by Ryan late on.
The missed chances, which also included a late Messi effort, mattered little in the end, with Argentina securing their spot in the quarter-finals of the competition courtesy of a narrow victory.
The South Americans had to survive a late scare, though, with Emiliano Martinez flying out to deny Garang Kuol, who is set to make the move to Newcastle United in January.