Bendtner had the chance to double his side's lead when he latched onto a delivery from Christian Eriksen, but he could only fire his effort over the bar from close range.
Denmark had the opportunity to double their lead when Pierre-Emile Hojbjerg was brought down in the box by Matija Nastasic and a penalty was awarded. Daniel Agger took the spot kick, but his strike was saved by Stojkovic.
In the second half, Serbia could have equalised as Branislav Ivanovic latched onto a cross into the box and met the ball with a powerful header, but his effort went wide of the target.
Adam Ljalic threatened for the visitors as he collected a pass from Lazar Markovic following good work from the Liverpool man, and his strike travelled towards the bottom corner, but Kasper Schmeichel was on hand to make the stop.
Serbia continued to pose problems for the home side and Aleksandar Mitrovic should have found a leveller for his team, but he could only head Ivanovic's cross over the bar from close range.
The visitors were to rue their missed chances as Denmark secured their victory in the 87th minute when Jakob Poulsen was teed up by Bendtner on the edge of the box, and he drilled his left-footed strike into the corner.
The result leaves Denmark three points clear of second-placed Albania, with five matches to play in their qualifying campaign.