The hosts started the match brightly and had territorial advantage as well as better ball possession, with Charlie Adam in the thick of things.
The visitors were put into trouble again a minute later when Steven Naismith failed to find a proper touch with his head to a fine delivery.
However, Serbia soon came into their own and started to create chances, as the pace of the match became relentless.
The final chance of the first half fell to Serbia on the stroke of half-time, when Darko Lazovic forced Scotland goalkeeper Allan McGregor with a shot from 18 yards.
Scotland and Serbia continued to exchange chances in the second half, with the home team coming close from a corner on 50 minutes.
Christophe Berra headed Adam's corner from close range but failed to keep it on target.
Four minutes later Serbia's Aleksandar Ignjovski took advantage of a defensive slip-up, but shot wide off the target from the edge of the Scotland penalty area.
Kenny Miller then tried his luck in the 56th minute but his effort was kept out by Stojkovic.
Naismith could have broken the deadlock in the 65th minute when he was played in by Gary Caldwell, but the Everton attacking player flicked the ball just yards off the mark.
Both Serbia and Scotland had good chances in injury time, but failure to take them meant that their first Group A game ended in a goalless draw.