The Danes transferred up to 12 points alongside team leaders Ireland, who drew 0-0 away to Georgia, while Switzerland are just four points off the pace but with a match in hand.
Mick McCarthy’s Ireland could not set any momentum earlier in the day at Tbilisi and so were left frustrated with wayward finishing in their few chances to break the deadlock.
Romania took over second place in Group F after a 3-0 triumph at the Faroe Islands ensured by second-half targets from George Puscas, Ionut Mitrita, and Claudiu Keseru.
Sweden could move back over Romania with success in Malta afterward although defeat could open the doorway for three-time European champions Spain to secure eligibility with a win off to Norway.
Italy, that have six wins from six in Group J, can clinch their place at the forefront with victory over Greece in Rome.
Bosnia and Herzegovina reignited their eligibility hopes with a 4-1 triumph over second-placed Finland.
A brace from Juventus midfielder Miralem Pjanic and targets out of Izzet Hajrovic and Armin Hodzic pushed Bosnia to within 2 points of Finland, which might be united by Armenia should they defeat Liechtenstein in Vaduz.