The Danes, who have four points from their opening two games, know that qualification for the knockouts will be assured if they can avoid defeat in Moscow on Tuesday to a France side who have already booked their place in the second round.
Yet their performance in their 1-1 draw against Australia at the Samara Arena on Thursday gave some cause for concern to Hareide, whose side took an early lead through a stunning Christian Eriksen strike before being pegged back.
Mile Jedinak's penalty for Australia towards the end of the first half not only levelled the scores but also shifted momentum towards the Socceroos, who began the game under the cosh but ended firmly on the front foot.
Other than Eriksen, the Danes looked workmanlike in midfield.
They frequently ceded possession and invited Australia attacks, leaving Hareide to ponder where improvements can be made.
"We lost too many balls in midfield and that caused us problems," the coach said.
"We need to get better with our passing and get more rhythm in our play. Many of our choices today were not good enough and that was the problem."
Denmark came into the encounter boasting a watertight defence that had not conceded in their five previous matches, but any sense of invulnerability was shattered the minute Australia drew level through Jedinak's spotkick.
From that moment on the Socceroos, who have now failed to win any of their last five World Cup matches, looked the more likely victors, capitalising on Danish mistakes to pile on the pressure.
"We lost too many balls and we had to run too much so our game was restless. You know what? The World Cup is tough," said Hareide.
"I think they produced chances when we made mistakes. We lost the ball in the last third and they counter attacked on us.
"It was a tight match... We are very happy to have four points."
(Reporting by Toby Davis; Editing by Amlan Chakraborty)
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