The Croatian dominated the slalom leg of the event and held off the challenge of the speed specialists in the downhill for victory.
Renate Goetschl of Austria finished in second, with Germany's Martina Ertl in third.
Despite holding a 1.10 second cushion over the rest of the field after the two slalom runs, Kostelic had not been overly confident of holding of her challengers on the downhill run.
However, the margin proved more than enough, despite the best efforts of the the pre-race favourite Goetschl.
Lying fourth after the slalom, the downhill bronze medallist put in a supreme effort in posting the best time of 1:15.27.
Although it was enough to move her up from fourth to second, she was unable to knock Kostelic from top spot.
The 20-year-old, who has recovered from major knee surgery, followed the Austrian out of the starting gate and held on to win by 1.49 seconds.
"I still can't believe I won - I always dreamed about this," she said.
"Maybe when I sleep over then I will know that I won the Olympic medal," she added.
The gold is Croatia's first of any colour in a skiing event at the Winter Olympics.
For Goetschl, at 26 still in search of her first Olympic gold, the silver will come as little consolation as the mighty Austrian team expects nothing less than first place atop the podium.
But starting the downhill 2.22 seconds adrift of Kostelic she never really stood a chance.
Great Britain's Chemmy Alcott finished 14th, 8.06 seconds behind Kostelic.
Chemmy, 19, said: "I felt under no pressure at all. Being a British skier we always have the ability to shock with better-than-expected results."
British Ski and Snowboard Federation director Fiona McNeilly said: "I was hoping for anything in the top 15 from Chemmy so I'm delighted with her performance.
"We were a little worried about her slalom. This was the first time in seven months she has finished a slalom race.
"Chemmy is still planning to do all five events and will next be in action super-G on Sunday."