Australia had only ever reached the last eight at the World Cup once before, in 2015, losing their four other knockout matches across the years.
They started slowly against Denmark but Stadium Australia in Sydney erupted after Fowler’s great turn and pass set up Foord’s opener on the break.
Encouraged by their goal the Matildas dominated the rest of the half with another Foord effort deflecting off Denmark defender Rikke Sevecke and dropping narrowly wide.
As the Danes pushed after the restart Foord broke down the left and played the ball to Van Egmond, who teed up Real Madrid’s Raso to calmly slot home her third goal of the tournament.
With the game seemingly wrapped up Australia head coach Tony Gustavsson brought on captain Sam Kerr who was given a standing ovation when she replaced Raso with just over 10 minutes remaining.
The Matildas’ record scorer played her first minutes of the tournament having picked up a calf injury in training on the eve of the competition.
And she shone in the closing moments, driving down the right and thundering a shot over the bar.Denmark, playing at their first World Cup since 2007, were making their first appearance in the last 16 for 28 years.
They had the best of the opening exchanges with former Chelsea midfielder Pernille Harder shooting straight at Australia keeper Mackenzie Arnold when well positioned in the 16th minute.
The two-time quarter-finalists had more possession than their opponents but created very few clear chances after falling behind.
Harder volleyed wide in the final minutes but by then Denmark’s quarter-final chances had faded away.
Australia will discover their last eight opponents when France take on Morocco on Tuesday (12:00 BST).