Norway's Kjetil Jansrud, a bronze medallist in the showpiece downhill last weekend, topped the time charts in one minute, 53.24 seconds, although he has no form in the slalom, which will decide the podium places later.
Kostelic, a three-times Olympic silver medallist and a specialist in the turning disciplines, was 0.93 seconds down in seventh place and will be confident of moving up through the field on a course set by his father, Ante.
Lurking in fourth place, 0.61 seconds behind, is Norway's Alexander Aamodt Kilde.
Big French hope Alexis Pinturault was only 23rd quickest, 2.44 seconds back, although he could slice through the field if he hits top form in the slalom.
Svindal was sixth, 0.70 seconds behind, although his hopes of winning a medal in the only discipline missing from his Olympic portfolio will depend on whether he can find some pace through the poles in his weakest event.
"In combined, the slalom is two thirds so it's only one third done," the reigning Olympic super-G champion told reporters.
On another dazzling day in the mountains above the Black Sea coast, reigning super combined champion Bode Miller produced an erratic run down the shortened downhill course, running wide to ruin what looked like being a fast time.
The American was 12th, 1.43 seconds behind Jansrud, while team mate and super combined world champion Ted Ligety will have to find 1.93 seconds to get among the medals.
Miller was again critical of the sun-softened course, despite organisers moving the start time to 10am.
"If they started at 11 everybody has crappy snow more or less," Miller, the 24th man down, told reporters. "But it's always in hindsight you can make those calls.
"You had to take maximum risk with our position to try to get something out of it but unfortunately when the course is run like this, maximum risk really is risky."
Ligety said all was not lost.
"Two seconds is not that much time to make up on downhill skiers," Ligety said. "A second on Ivica is tough for sure but slalom, there can be huge swings really quickly."
Jansrud tipped Pinturault for a medal charge.
"He beat me by four seconds in the last combined," he told reporters after sweeping down the glistening 3.2km course 0.14 seconds quicker than Czech Ondrej Bank and 0.37 ahead of Austria's new Olympic downhill champion Matthias Mayer.
"I'll have to fight them all off, it's going to be tough but I'll do my best," he added.