The award was presented at a ceremony in the Croatian Parliament on Monday.
The European Citizen's Prize was launched by the European Parliament in 2008 to recognise exceptional achievements by Europeans. Candidates are proposed by members of the European Parliament and winners are awarded with a symbolic prize in the form of a medal of honour.
Barun and the Croatian Transplant Association received an identical number of votes.
"Today is not a typical day in the Croatian Parliament. Today is the day when we celebrate mankind, love, unity and solidarity. I wish there were more days like this. It would give me great joy to see the Croatian Parliament permanently taking part in events like this," Parliament Speaker Gordan Jandrokovic said at the award ceremony.
The award on behalf of the Croatian Transplant Association was received by Tanja Watz who dedicated it to the families of organ donors.
"Life does not necessarily end with organ failure, 400 organ transplants in Croatia each year are the best proof of that," Watz said.
The central ceremony for award recipients will be held in the European Parliament in Brussels on 11 October.