The equation requiring India to win both the reverse singles to stay in World Group I, Nagal put a brave fight but lost 5-7 3-6 in one hour and 37 minutes in the first reverse singles.
That defeat gave Denmark an unassailable 1-3 lead and pushed India to World Group II.
Prajnesh Gunneswaran then took court in the dead rubber against Elmer Moeller and won 6-4 7-6(1) in one hour and 39 minutes.
On Friday, Nagal had erased a one-set deficit to pip August Holmgren 4-6 6-3 6-4 in two hours and 27 minutes to make it 1-1 after Bhambri lost the opening singles 2-6 2-6 to the rising teen sensation Rune.
Bhambri has quit singles and now plays only doubles on tour.