Yussuf Poulsen, who conceded the penalty earlier on, became the second half savoir by slotting home what turned out to be the winning goal.
Danish players speak to the referee after VAR is used to award a penalty to Australia.
After their opening defeat by France, Australia coach Bert van Marwijk resisted the temptation to draft in attacking reinforcements, leaving veteran forward Tim Cahill on the bench for the second match in a row.
The Socceroos again outperformed expectation by dominating a Denmark team ranked 12th in the world for long periods, but their failure to find a victor leaves them with just one point to France' six and Denmark's four.
The Danes were initially headed for victory at the Samara Arena courtesy of Christian Eriksen's fine strike after seven minutes, but Mile Jedinak converted from the spot eight minutes after the half-hour mark to earn Australia a share of the spoils.
It never came, leaving the Socceroos to depart the raucous stadium to a standing ovation but just one point from their two games.
Eriksen's free kick caused mayhem in the box just minutes later, with Mat Ryan collecting in relief after the ball came off the leg of Trent Sainsbury a yard from the goal-line. The team is representing Australia on the world's biggest sporting stage and millions of football fans around the country are getting right behind them.
After Denmark's Pione Sisto put a good opportunity wide, Australia gradually got on top, but the nearest they came to a victor was when Leckie's cross was turned over his own goal by Henrik Dalsgaard and Mooy blasted over from distance.
For the second game running the Aussies were bold and dominant but failed to reap their just rewards, leaving them with only one point in their World Cup campaign to date - but with a slight chance of qualifying for the knockout rounds on the final matchday. However, they did give a lot of unsafe set- pieces away for France, and Denmark will be looking to capitalise on this, where the expertise of Christian Eriksen can come into play.
The Australian midfield of Mile Jedinak and Aaron Mooy is solid, but they need to make something happen up front for the likes of Andrew Nabbout and Tom Rogic to finish the buil-up play.
"They were afraid of us", Van Marwijk told reporters following the game, via Fox Sports.
"I feel like I'm ready", said Arzani, who scored his first goal in a 2-1 friendly win against Hungary last month.
"We played very well and created chances and it is an unbelievable pity that we didn't score".
"From the first day the team has improved every day and they surprised me today again". "You know what? The World Cup is tough".