Bernard Foley's controversial last-gasp penalty broke Scotland hearts as Australia edged a Twickenham thriller 35-34 to set up a Rugby World Cup semi-final against Argentina.

Australia were overwhelming favourites to beat Vern Cotter's side in the last eight, but the Wallabies were given a major scare and won for the first time in a World Cup game in which they had trailed at half-time.

Mark Bennett's try six minutes from time looked to have given Scotland a sensational victory, but they were denied a place in the last four for the first time in 24 years when Foley held his nerve with the boot, Cotter's men having been punished for an accidental offside.

Plucky Scotland led 16-15 at the break courtesy of a Peter Horne try and 11 points from the boot of the tournament's top scorer Greig Laidlaw, who scored 19 in total, with Adam Ashley-Cooper, Drew Mitchell and Michael Hooper crossing for the Wallabies.

Mitchell went over again after Sean Maitland was yellow-carded just after the break, but tries from Tommy Seymour and Bennett put Scotland 34-32 in front with time running out after Tevita Kuridrani had gone over at the other end.

There was one final twist though as Foley ensured there will be no Northern Hemisphere team in the semi-finals and left Scotland shell-shocked. Many observers felt South African referee Craig Joubert should have referred to the television match official in determining whether to award a penalty or a scrum to Australia in the final, game-changing moments. 

Australia were in front after only eight minutes in West London, when Kuridrani breached the Scotland defence and his pass outside allowed Ashley-Cooper to go over in the corner. 

Foley failed to convert and Laidlaw's penalty cut the deficit to two points before a fired-up Scotland capitalised on some poor Wallabies defending to score a try of their own.

Horne was the benefactor when Australia went to sleep at a ruck and the centre casually picked the ball up and could hardly believe his luck as he went under the posts unchallenged.

Laidlaw added the extras and slotted another penalty to make it 13-5 after 21 minutes.

Australia responded and Mitchell scurried over after they worked the ball quickly from right to left, but Foley again was unable to convert and the clinical Laidlaw made it 16-10 from the tee.

Scotland were pegged back again just before half-time, though, when Australia executed the throw and drive with great effect to allow Hooper to touch down but another missed conversion from the out-of-sorts Foley ensured the Scots led at the break.

Mitchell then went over again in the corner just three minutes into the second half after Maitland had been sin-binned for a deliberate knock-on, with Foley this time successful with a tricky conversion to make it 22-16. 

A Laidlaw penalty cut the gap to three points and Scotland breathed a sigh of relief when Ashley-Cooper thought he had his second try, but was denied due to a knock-on at the back of the scrum before Foley stretched Australia's lead to 25-19 from the tee.

Foley was then at fault when his kick was charged down by Finn Russell, who offloaded to the onrushing Maitland and he raced over in the corner.

Kuridrani barged over for a converted try to put Australia 32-24 up, but Laidlaw slotted over another penalty and then put Cotter's men in front after Bennett intercepted a terrible pass from James Slipper to race away and go under the posts, only for Foley to have the final say in a cruel finish for the underdogs.