Serbia, who have one point from five games in a dismal qualifying campaign, dominated the hosts for the entire second half but a combination of last-ditch defending and some spectacular saves from keeper Kasper Schmeichel sent them home empty-handed.
Denmark took the lead in the 13th minute from a superb flowing move, Pierre Hojbjerg sweeping a long ball to Nicklas Bendtner whose deft touch placed it perfectly in the path of Poulsen who drove it home.
The Danes could have gone two up just after the half-hour when Matija Nastasic clumsily tripped Pierre Hojbjerg but Denmark captain Daniel Agger's penalty was easily pushed away by Vladimir Stojkovic.
Behind at the break, the Serbs launched a second-half onslaught but Schmeichel pulled off a superb save to palm Aleksandar Mitrovic's powerful header onto the crossbar and away to safety.
Schmeichel was soon called into action again, parrying a stinging shot from Adem Ljajic to keep the Danes' slender lead intact.
The Danes spent much of the second half clearing their lines, Agger putting in a commanding performance at the back, manhandling the Serbian forward line and stealing the ball off the toes of the Serbs on several occasions.
The sucker-punch came in the 88th minute when the Danes made a rare foray forward and substitute Jakob Poulsen held off Nikola Maksimovic to rifle the ball low past Stojkovic.
Portugal's 3-2 victory away to Armenia earlier in the day means they top Group I on 12 points, with the Danes second on 10.
Albania are third on seven points and a game in hand, with Serbia and Armenia rooted to the bottom with a single point each.
(Reporting by Philip O'Connor in Stockholm; editing by Justin Palmer)
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