Caitlin Foord's goal in the 29th minute, brilliantly teed up by Mary Fowler, sent the Matildas on their way before Hayley Raso added a second in the 70th minute in front of 75,784 screaming fans at Sydney's Stadium Australia.
Superstar captain Sam Kerr also returned off the bench late as Australia closed out a comprehensive, professional win.
"I'm very proud of the players' maturity tonight," said Gustavsson.
"Denmark, they surprised us and we knew we'd have problems between the midfield and the backline, but the maturity showed tonight.
"We found our way into the game and with some individual brilliance on the break that was enough to get away with the win tonight."
After originally being named on the bench as coach Tony Gustavsson stuck with an unchanged starting line-up, Kerr entered the field to a standing ovation in the 80th minute.
They were her first minutes of the tournament, after overcoming a calf injury suffered on the eve of Australia's opening game.
"Sam coming back now could be a good thing in the sense that we don't need to look to her to carry everything on her shoulders," Gustavsson said.
"It's a team effort and a team tournament and (having her back) is like the icing on the cake."
Australia will next play either France or Morocco, who face off in Adelaide on Tuesday night, at Brisbane's Suncorp Stadium on Saturday evening.
It is just their second time in the final eight of the World Cup, after previously reaching the marker in Canada in 2015.
The magnificent Foord gave Denmark right-back Janni Thomsen nightmares all night while Fowler's sublime passing range unlocked both goals.
Centre-back Alanna Kennedy was dominant in the air and consistently thwarted Danish attacks as the Matildas sealed a professional victory.
The Australians looked nervous early and were given some troubles by a determined Danish outfit.
Danish star Pernille Harder caused headaches and spurned chances after superb bursting runs in the 10th and 17th minutes.
Stand-in skipper Steph Catley went down and received treatment on her lower leg in the 18th minute.
The break gave the Matildas the breathing room they needed.
Nine minutes later, Australia took the lead.
Fowler cut past a defender in midfield, then neatly turned on the ball and switched onto her left foot, before launching a precise ball into the path of a charging Foord out wide.
Foord took a touch, steadied then buried a cool shot through goalkeeper Lene Christensen's legs before wheeling away to her home crowd, cupping her ear in celebration.
Australia never looked challenged from there and earned a deserved second goal after Fowler whipped a ball into Kyra Cooney-Cross on the left wing, then burst forward to receive it back.
Fowler slipped through to Emily van Egmond, who had the presence of mind to knock it back for Raso to drive home her third goal of the tournament..
When Kerr entered the fray with 10 minutes of regular time remaining, the skipper sent the crowd into raptures.
She had them on their feet when she burst forward and shot off target in the 87th minute, in an exciting sign of things yet to come.
"I feel good. I was a bit rusty," Kerr told Channel Seven.
"No, I felt great. Such relief. I'll sleep better tonight than I have the last three weeks."
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