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|Republic of Ireland's defender Shane Duffy (right) cannot reach a cross as Denmark's defender Simon Kjaer (centre) defends during the Group D Euro 2020 football qualification match. (Lorraine O'SULLIVAN/AFP)|
Switzerland also booked their place at Euro 2020 with a 6-1 win over Gibraltar to claim top spot in Group A.
Ireland still have a chance to play in the Euros on home soil with Dublin one of 12 host cities, but now need to come through two playoff matches in March as Matt Doherty's equaliser to cancel out Martin Braithwaite's opener for Denmark was too little, too late.
The hosts were seeking revenge for the 5-1 thrashing that sent Denmark to the 2018 World Cup.
However, in six meetings between the sides over the past two years Ireland have scored just three goals and a lack of cutting edge again thwarted Mick McCarthy's men.
The hosts looked the most likely to make the breakthrough in a first half of few clear chances.
Conor Hourihane had the best of them but fired too close to Kasper Schmeichel from a narrow angle as he insisted on taking the shot with his stronger left foot.
In contrast to recent meetings between the sides, it was Denmark who were happy to sit back and force Ireland to break them down, knowing a point would be enough for qualification.
Christian Eriksen was the major difference between the sides in the Danes' 5-1 victory in the Irish capital two years ago with a hat-trick.
But the Tottenham midfielder's struggles for form at club level had been replicated for his country this season and the visitors struggled to get their star man into the game.
Yet, even with Eriksen stifled, Denmark produced the moment of quality they needed 17 minutes from time when Henrik Dalsgaard's cross was met on the stretch by Braithwaite.
Irish hopes of avoiding the playoffs seemed to be gone, but Doherty's towering header at the back post from Enda Stevens's cross set up a grandstand finish.
The Denmark box was then bombarded with crosses with Irish captain Shane Duffy thrown forward from centre-back, but there was to be no dramatic late winner.
Switzerland made sure they beat the Danes to top spot in the group by hitting Gibraltar for six.
Arsenal midfielder Granit Xhaka was among the goalscorers as he put his woes at club level to one side after being stripped of the captaincy by Gunners' manager Unai Emery.
Euro 2020 qualifying results:
Republic of Ireland 1 Denmark 1
Gibraltar 1 Switzerland 6
Sweden 3 Faroe Islands 0
Spain 5 Romania 0
Malta 1 Norway 2
Liechtenstein 0 Bosnia and Herzegovina 3
Italy 9 Armenia 1
Greece 2 Finland 1