Sam Prendergast, who delivered the killer pass for the second try, converted both for 14-0.
However, Prendergast came off second best shortly afterwards as Len Ikitau got the better of him in going over to score a try.
James O’Connor, who broke off a holiday in Morocco after being called up by Schmidt, converted for 14-7.
Prendergast’s ability to do the unexpected is what head coach Andy Farrell likes about him and the 22-year-old delivered as the clock ticked towards 30 minutes.
His crossfield kick to Tommy O’Brien was perfect and although the wing was brought to ground, 27-year-old Hansen, who has only recently returned from a foot injury, was in the right place to pick the ball up and cross the line.
The Aussies, showing the same dogged spirit that Schmidt instilled in Ireland during his time in charge, kept snapping away at the hosts’ heels and struck on the stroke of halftime.
Flanker Fraser McReight bundled over after a period of persistent pressure – O’Connor added the extras to leave them just 19-14 down at the break.
Australian wing Harry Potter produced some magic in defence when he held up Irish hooker Dan Sheahan on the tryline as the hosts pressed in the early part of the second half.
Prendergast gave the Irish breathing space with a fabulous long range drop goal for 22-14 with 25 minutes remaining.
Prendergast looked to have set up a decisive match-winning try for O’Brien with a delightful crossfield kick but it was ruled out for a knock-on by James Lowe.
Prendergast threw the kicking tee away in frustration and departed after an outstanding performance.
His replacement, Jack Crowley made an instant impact, converting a penalty for 25-14 with under 20 minutes remaining.
The hosts dashed any outside hopes the Wallabies had of coming back when the irrepressible Hansen’s catch inside the Aussie 22 led eventually to captain Caelan Doris touching down.
Though the visitors were reduced to 14 men with Nick Frost sin-binned, they showed great spirit as Billy Pollard went over for 32-19.
However, the Irish were to have the last laugh as Ryan Baird and then Robbie Henshaw went over – Crowley converting both – to round off a hugely satisfying night for the hosts.