"This act has inflicted immeasurable damage on the reputation of Croatian citizens and their homeland all over the world. Therefore, we must finally put a stop to such things," she said in a statement released on Saturday.
Croatia played Italy behind closed doors in Split having been handed a one-match ban for racist chants by their fans in a 5-1 home win over Norway in Zagreb in March.
That followed incidents during a 1-1 draw against Italy in Milan last November when flares were hurled on to the pitch.
The swastika sign imposed in the grass at Split's Poljud stadium through a chemical agent days before the kickoff became visible during the match.
"I condemn in the strongest possible terms the swastika on the pitch. I demand an urgent investigation and a decisive response from the relevant institutions to find and process the perpetrators," Grabar-Kitarovic said.
The swastika incident left Croatia players and the country's soccer ruling body, the HNS, fearing drastic punishment from the UEFA after the result left them top of Group H with 14 points from six games, two ahead of Italy and four clear of Norway.
"This is a clear attack on football and we will without a doubt be severely punished," HNS secretary-general Damir Vrbanovic told a news conference on Saturday.
"We have all been slapped in the face by hooligans and it was not an accident, it was a calculated move. The cameras will identify the perpetrators, but the shame will stay," he said.
(Reporting by Zoran Radosavljevic, editing by Ed Osmond)
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