Croatia won 12-11 in the high scoring game after the two Balkan arch rivals had traded the lead in a gruelling yet spectacular encounter.
Serbia came into the tournament as defending World, Olympic and European champions with the first two of those titles coming after finals against Croatia.
Yet three big saves by goalkeeper Marko Bijac allowed the Croats to end a losing streak that stretched back to 2010.
"We weren't able cope with the atmosphere tonight and we didn't get that right," Serbia's Milos Cuk said. "What can you do? That's sport and Croatia were better tonight. We'll fight for sure for the bronze."
Hungary beat Greece 7-5 in an intense, but far more defence-oriented match.
The Hungarians enjoyed a measure of luck as four powerful Greek shots hit the goal post in the third quarter.
Hungarian head coach Tamas Marcz heaped praise on their defence.
"We played an all-around huge defensive game," Marcz told reporters. "(Goalkeeper) Viktor (Nagy) was downright fantastic."
(Additional reporting by Rod Gilmour; Editing by Toby Davis)
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