Darren Randolph made fine saves from Christian Eriksen and Yussuf Poulsen either side of the break, but Denmark wasted a series of golden opportunities to open the scoring before Hojbjerg belatedly did so, and they were ultimately made to pay.

The Danes entertain Georgia on Monday evening with just two points from the opening two qualifiers, while Ireland have banked seven ahead of Gibraltar's trip to Dublin the same night.

Irish obstinacy – three of the four most recent encounters between the sides have finished 0-0 – has been a source of great exasperation to the Danes, with midfielder Thomas Delaney describing the Republic as his nation's "most annoying" opponent in the run-up to the game.

Quite what the mood was in the home dressing room at the break on this occasion was a matter for debate with Denmark dominating the first half for long periods – Eriksen showed few signs of a Champions League final hangover – but without finding a way past goalkeeper Randolph.

It was Erisken who tested Randolph with an 11th-minute free-kick which bounced nastily in front of the goalkeeper, although he should have done better with 27 minutes played when he took advantage of a collision between Glenn Whelan and James McClean to race clear, although his decision to cut back on to his right foot to shoot allowed Jeff Hendrick to make a timely intervention.

In the meantime, Martin Braithwaite spurned two passable opportunities, but Kasper Schmeichel had found himself desperately clawing away Duffy's improvised effort after Conor Hourihane had caught out the home defence with a devilish free-kick.

The Republic gradually managed to stem the tide as the half-time whistle approached, but McCarthy had work to do at the break.

Poulsen squandered a glorious opportunity to put the Danes ahead within eight minutes of the restart when, after running on to Nicolai Jorgensen's perfectly-weighted pass, he stepped inside Duffy's lunge, but could not find a way past Randolph.

However, Ireland responded within seconds when Robbie Brady picked out McClean inside the box and he drilled a shot into the near post, where Schmeichel made a vital block.

David McGoldrick flicked a header onto the roof of the net with the visitors sensing an opportunity, but they were quickly put back in their place when Poulsen's 59th-minute run only failed to result in the opening goal when Hourihane's desperate challenge sent the ball agonisingly wide of the post.

Braithwaite and Jorgensen both passed up good chances as McCarthy's men became increasingly stretched, but the deadlock was broken with 14 minutes remaining when Hojbjerg timed his run to perfection to get to Jens Larsen's cross ahead of Richard Keogh and glance a header past Randolph.

The lead lasted just nine minutes, however, and Ireland snatched a point as the game approached its final throes when Duffy climbed high above Simon Kjaer to power home a header from substitute Alan Judge's free-kick and secure a potentially priceless point.

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