Joakim Maehle scored in the 53rd minute to send the Danes to Qatar as they build on their surprise run at Euro 2020 this summer when they reached the semifinals.
England sputtered as a combative Hungarian side held it to a 1-1 draw while Portugal thrashed Luxembourg.
Sweden stayed two points ahead of Spain in Group B thanks to a 2-0 win against Greece.
The results: Group A: Portugal 5 (Ronaldo 8-pen, 13-pen, 87, Fernandes 18, Joao Palhina 69) bt Luxembourg 0; Serbia 3 (Vlahovic 30-pen, 53, Tadic 83-pen) bt Azerbaijan 1 (Makhmudov 45+2).
Group B: Kosovo 1 (Muriqi 45-pen) lost to Georgia 2 (Okriashvili 11, Davitashvili 82); Sweden 2 (Forsberg 59-pen, Isak 69) bt Greece 0.
Group C: Bulgaria 2 (Nedelev 53, 63) bt Northern Ireland 1 (Washington 35); Lithuania 0 lost to Switzerland 4 (Embolo 31, 45, Steffen 42, Gavranovic 90+4).
Group D: Kazakhstan 0 lost to Finland 2 (Pukki 45, 48); Ukraine 1 (Yarmolenko 15) drew with Bosnia and Herzegovina 1 (Ahmedhodzic 77).
Group F: Denmark 1 (Maehle 53) bt Austria 0; Faroe Islands 0 lost to Scotland 1 (Dykes 86); Israel 2 (Zahavi 28, Dabbur 49) bt Moldova 1 (Nicolaescu 90+5).
Group I: Albania 0 lost to Poland 1 (Swiderski 77); England 1 (Stones 37) dreww with Hungary 1 (Sallai 24-pen); San Marino 0 lost to Andorra 3 (Pujol 10, Moreno 53, Fernandez 89).
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