Denmark defender Joakim Maehle netted the game’s only goal in the 53rd minute, finding space inside the box before drilling a shot in at the near post.
The Danes follow Germany in qualifying for Qatar 2022, with Die Mannschaft confirming their place on Monday with a 4-0 win at North Macedonia.
Denmark’s Tottenham midfielder Pierre-Emile Hojbjerg told Kanal 5: “You can only dream about things like this. It’s crazy.
“This is huge for me, it’s huge for the team, for Danish football and for Denmark.
“With the age and quality we have, we will only keep growing. It’s about staying humble and keeping our minds in the right place.”
The triumph marks Denmark’s continued growth since Christian Eriksen’s collapse on-field during their Euro 2020 opener when they lost to Finland. Eriksen remains unavailable as he recovers from cardiac arrest.
Sweden triumphed 2-0 over 10-man Greece to remain on Spain’s tail in Group B, while Scotland edged Faroe Islands 1-0 to stay second in Group F and Switzerland moved level on points with Italy in Group C after winning 4-0 at Lithuania after a Breel Embolo double.