Australia outscored Scotland five tries to three to ensure it will be a set of all-Southern Hemisphere semi-finals next weekend after a 35-34 win at Twickenham on Sunday.
Scotland were outstanding and went desperately close as Mark Bennett's interception score looked to have clinched an enormous upset, after opportunist tries from Peter Horne and Tommy Seymour kept them in the hunt.
But the Wallabies' attack was too dangerous as Drew Mitchell crossed twice along with Adam Ashley-Cooper, Tevita Kuridrani and Michael Hooper, before Bernard Foley snatched back the lead in the 78th minute with a penalty to win the game.
If Australia play with this much width and quick ball at the breakdown next week, then their semi-final with Argentina could be a classic.
Never has Twickenham been so loud for the men in blue from north of the border. Vern Cotter's team were willed on by the top half of the earth to make sure a northern hemisphere side made the final four, but now no sides from the north are left in the competition.
The Six Nations wooden-spooners though are undeniably on an upward curve and while this was a step too far for now, they should not be too downhearted after leaving everything on the pitch.
Every time Australia entered the Scottish 22 they looked certain to score and the number of chances that went begging were minimal.
However even with that try-scoring threat they were never comfortably out of reach, as Scotland doggedly battled away, backed by a roaring Twickenham crowd.
The Wallabies' ability to recycle the ball at the ruck quicker than Will Genia can click his fingers has made them a nightmare to contain throughout this tournament, and it was the same again early on.
Scotland's defence was put to work in the opening exchanges, with Mitchell unable to gather an inside pass from Foley from two metres out, but it wasn't long before the Wallabies crossed the whitewash.
Tommy Seymour burst out of the line to try and stop Kuridrani early only to miss his tackle, allowing the big centre through to release Ashley-Cooper into the space that Seymour had left behind.
Scotland though are not a side who mope on mistakes and they deservedly scored three points through a Laidlaw penalty after a patient attack inside Australia's 22, making it 5-3.
Peter Horne's try soon followed, the centre smartly spotting no Wallaby tacklers behind the ruck before scooping the ball up and darting behind the defence from five metres out. Laidlaw's conversion gave them a 10-5 lead.
Australia's scrum has been the talk of the World Cup but it buckled to give Laidlaw a penalty shot from 47 metres out, the Scotland captain making no mistake as he side extended their lead.
The Wallabies remained eager to go wide and a penalty for hands in the ruck, despite Sean Maitland's smart cover tackle, set Australia up in Scotland's 22 from which they eventually scored in the corner through Mitchell after sucking in the Scottish defence. Foley again missed the conversion from out wide.
Another penalty though against the Australian scrum, after Scott Sio collapsed, allowed Laidlaw another shot at the posts to make it 16-10.
Runs from Maitland, Mark Bennett and Blair Cowan meant Scotland were always asking questions of Australia, but back-to-back penalties set the Wallabies up for a driving maul five metres out for Michael Hooper to score the third try.
Again Foley missed going for the extra two, leaving him zero from three by half-time as Scotland went in ahead 16-15.
Scotland needed everything to go their way to pull off a shock, so an extremely harsh yellow card on Maitland was the worst possible start.
After the success of their maul earlier on the Wallabies understandably returned to the corner, except this time Genia darted down the blindside and released Mitchell for his second try in the corner.
A fourth Laidlaw penalty kept Scotland right in the contest at 22-19 but a huge loss of composure from the restart led to Seymour being bundled into touch five metres from his own line.
It appeared that Ashley-Cooper had matched Mitchell with a second try of his own after the Wallabies whipped the ball wide, but a small knock-on from Genia at the ruck brought the play back to keep Scotland alive.
Foley's first penalty after 52 minutes gave Australia a six-point cushion as the error counts rose on either side, with Foley committing the biggest of all.
An undercooked chip kick was charged down by Finn Russell and despite Genia tracking back to bring the fly-half down, his offload off the ground released Seymour to score as Twickenham erupted. Laidlaw however couldn't add the crucial conversion, leaving Australia ahead 25-24.
Kuridrani responded for Australia after a timely steal at the lineout from Rob Simmons had Scotland on the rack again before the outside centre powered his way over.
Scotland were now more than a score behind but pressed on, adding another Laidlaw penalty to make it 32-27 heading into the final ten minutes.
Bennett looked to have snatched a famous win with a well-taken interception that sent the crowd crazy but a fiercely debated call for accidental offside deep in Scottish territory set Foley up with a penalty for the win.
He didn't miss, ending the hopes of Scotland and those north of the equator. Heartbreak for Cotter's side. The Wallabies slip through.
Man of the Match: Greig Laidlaw, Richie Gray and Mark Bennett deserve huge credit but Adam Ashley-Cooper was a massive threat throughout.
Moment of the Match: If Twickenham had a roof it would be blown off when Mark Bennett broke away for the line and a huge upset looked on the cards.
Villain of the Match: An absurd call for a deliberate knock-on saw Sean Maitland yellow carded by TMO Ben Skeen, with Drew Mitchell then scoring in his corner.
Tries: Ashley-Cooper, Mitchell 2, Hooper, Kuridrani
Cons: Foley 2
Pens: Foley 2
Tries: Horne, Seymour, Bennett
Cons: Laidlaw 2
Pens: Laidlaw 5
Yellow Card: Maitland
Australia: 15 Kurtley Beale, 14 Adam Ashley-Cooper, 13 Tevita Kuridrani, 12 Matt Giteau, 11 Drew Mitchell, 10 Bernard Foley, 9 Will Genia, 8 Ben McCalman, 7 Michael Hooper, 6 Scott Fardy, 5 Rob Simmons, 4 Kane Douglas, 3 Sekope Kepu, 2 Stephen Moore (c) 1 Scott Sio
Replacements: 16 Tatafu Polota-Nau, 17 James Slipper, 18 Greg Holmes, 19 Dean Mumm, 20 Sean McMahon, 21 Nick Phipps, 22 Matt Toomua, 23 Quade Cooper
Scotland: 15 Stuart Hogg, 14 Sean Maitland, 13 Mark Bennett, 12 Peter Horne, 11 Tommy Seymour, 10 Finn Russell, 9 Greig Laidlaw (c), 8 Dave Denton, 7 John Hardie, 6 Blair Cowan, 5 Richie Gray, 4 Tim Swinson, 3 WP Nel, 2 Fraser Brown, 1 Alasdair Dickinson.
Replacements: 16 Kevin Bryce, 17 Gordon Reid, 18 Jon Welsh, 19 Alasdair Strokosch, 20 Josh Strauss, 21 Henry Pyrgos, 22 Richie Vernon, 23 Sean Lamont.
Referee: Craig Joubert (South Africa)
Assistant Referees: Glen Jackson (New Zealand), Pascal Gauzere (France)
TMO: Ben Skeen (New Zealand)