The deal divided Bosnia into two entities - a Serb republic and Muslim-Croat federation.
President Kostunica's comments came after his visit to Sarajevo - the first by a Yugoslav leader since Bosnia broke away from Yugoslavia in 1992, leading to a brutal war which raged until 1995.
He said that he would respect the Dayton agreement, which put an end to the Bosnian war, and was signed by his predecessor Slobodan Milosevic.
The deal divided Bosnia into two entities - a Serb republic and Muslim-Croat federation.
However, Mr Kostunica refused to apologise for the war, saying he would not use empty words.
The visit was arranged on Saturday in a bid to calm Bosnian authorities angered by Mr Kostunica's original plans to only visit the Bosnian Serb Republic, where he attended the reburial of a Serb poet, Jovan Ducic.
'Historic and symbolic'
The United Nations administrator in Bosnia Jacques Klein, who also participated in the meeting, described the visit as historic and symbolic.
He said it showed that Mr Kostunica recognised that the relationship between Bosnia and Yugoslavia was the key to stability.
Mr Kostunica met representatives of Bosnia's three-member presidency - Serbian chairman Zivko Radisic, Muslim member Halid Genjac and Foreign Minister Jadranko Prlic.
An aide represented the Croat member of the presidency who was unable to attend.
Mr Kostunica later travelled to Yugoslavia's smaller republic, Montenegro, where he held talks on forming a new federal government with Zoran Zizic of the Socialist People's Party, who is likely to become the prime minister of Yugoslavia.
War tribunal
Although Mr Kostunica said he was a very firm defender of the Dayton accord, he has been critical in the past of the war crimes tribunal in The Hague, set up under the accord.
He has called the tribunal a "political institution" and has refused to hand over Mr Milosevic, who has been indicted for war crimes.
Mr Kostunica said Mr Milosevic was no longer playing an active role in politics.
Mr Milosevic is widely blamed for inciting the conflict and supporting the Bosnian Serb separatists.
Thousands were slaughtered in the "ethnic cleansing" that marked the conflict and nearly 20,000 people are still unaccounted for.