Zigic's opposite number Milivoje Novakovic gave Slovenia a 63rd minute lead in a match littered with fouls, after both sides had missed several chances in the first half.
The result left Serbia on four points from two games after their opening 3-0 win in the Faroe Islands, while Slovenia have one following their 1-0 home defeat by Northern Ireland.
Slovenia striker Zlatko Dedic twice went close in the first half after their goalkeeper Samir Handanovic parried a fierce shot by Zdravko Kuzmanovic at the other end.
Nemanja Vidic denied Novakovic with a desperate block shortly after the break but Slovenia's hitman delighted the 3,000 away fans when he superbly turned in a looped through ball by winger Andraz Kirm.
Serbia midfielder Zdravko Kuzmanovic had a shot cleared off the line before Zigic rose abobe his marker to head home an inviting cross by right-back Antonio Rukavina.
"We have to be content with the result because we equalised so late in the game, they caused us a lot of problems and our final pass was poor all evening," Serbia's stand-in coach Aleksandar Jankovic said.
"We have to raise our game for our upcoming matches against Estonia and Italy if we are to stay in contention for a berth in the finals," he said.
Jankovic is in charge during head coach Radomir Antic's four-game touchline ban, imposed on him by FIFA for insulting the referee in Serbia's 2-1 World Cup defeat by Australia.
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