The Wallabies needed all of their nous in both attack and defence to break down a stubborn Argentina side 29-15 at Twickenham.
Winger Adam Ashley-Cooper was the hero, scoring a hat-trick of tries, while lock Rob Simmons also crossed the whitewash early on and fly-half Bernard Foley notched nine points with the boot.
All of the Pumas' points came from number 10 Nicolas Sanchez but, despite enjoying plenty of possession, the Pumas could not burst through the ferocious Aussie defence.
It means the two-time champions will now face their great rivals New Zealand in Saturday's final, while Argentina will have to settle for a place in the Bronze medal match against South Africa the previous evening.
Despite the gargantuan nature of his team's achievement, Cheika was in typically downbeat mood after the match, insisting they have a number of areas they need to work on if they are to beat the All Blacks.
"What we did was let the game open up which suited them," he is quoted as saying by the BBC. "I'm really happy with the lads' defence. We had to work hard to stop them from scoring a try.
"We can improve again and that's really important now. Now tonight we can think we've given ourselves a chance to play in a final and we will work out how we will do it against a dynamic team."