The pro-European businessman Petro Poroshenko has won a landmark presidential election in Ukraine with 56% of the vote, according to exit polls, clearing the 50% threshold to win outright without a second round. Former prime minister Yulia Tymoshenko was trailing far behind, with about 13%.

Ukrainians flocked to the polling stations on Sunday in what was seen as the most important election since independence. Millions of citizens in the restive east, however, did not vote at all, either because of separatist sympathies, feelings of intimidation by pro-Russian militias or simply a lack of polling stations.

"Today we can definitely say all of Ukraine has voted, this is a national vote," said Poroshenko from his campaign headquarters shortly after the exit polls were released. "The first steps that we will take at beginning of presidential office should be focused on stopping the war, to put an end to this chaos and bring peace to a united Ukraine." Poroshenko promised to hold parliamentary elections before the end of the year. "When there is a parliamentarian crisis, the only solution in a democracy is early elections," he said.

He also committed himself to start a dialogue with Russia and launch negotiations for a new treaty to replace the Budapest memorandum – the 1994 document in which Russia, the US and the UK promised to protect Ukrainian territorial integrity in exchange for the country giving up its nuclear arms. His first trip as president, he said, would be to the Donbass region.

Earlier in the day, voters said they felt the election was an important step toward solving the country's political crisis, and several said they wanted Poroshenko to exceed the 50% threshold to avoid a runoff vote in three weeks.

"Since Russia doesn't recognise our government, it's very important that the people say that now there is one person they support. Then the whole world will understand that their position is absurd," said Vladimir Pestenkov, an executive at an IT company.

But truck driver Alexander Pivin was one of a significant minority sceptical about Poroshenko. He voted for a controversial radical, Oleh Lyashko, the only candidate who went to the restive regions in the east, where he took part in operations against separatists.

"I don't like that they're forcing Poroshenko on us as the unity candidate," Pivin said. "At this moment, when people are dying in the east, politicians shouldn't be here [Kiev] or in the west where it's peaceful, they should be at the hotspots."

Lyashko came third, with 8% of the vote, according to exit polls. Turnout was reported to be high in most of the country.

"The turnout is a lot higher this time, which is good, although the election workers are barely able to keep up," said Olesya Maximenko, a vote observer with the civil society non-governmental organisation OPORA in Kiev, said turnout was high. "These elections cost us lives and blood so, knowing the price, the least people could do is come out and vote."

In the east, polling day revealed how much work the government in Kiev has in store to bring the region back under control. It was always expected that in separatist strongholds such as Slavyansk there would be no voting but, more surprisingly, in Donetsk, a city of close to 1m people, not a single polling station opened. Even in the morning, sources inside the pro-Kiev administration said they hoped to have a number of polling stations open by the afternoon, but that did not happen.

In Dokuchayevsk, several of the 13 polling stations had planned to open but in the end none did. At School Number Three, there were plans for a late opening at 10am after the local committee finally received ballot papers overnight but, just before voting opened, a separatist representative arrived and demanded that the station was closed.

"He asked politely, but made it clear that if we did not accede, he would come back," said one of the election officials. The separatist made off with the ballot papers, and the town's only remaining polling station closed.

The closest town to normality in the region was Mariupol, scene of violence on 9 May when pro-Ukrainian forces entered the town and clashes broke out in which unarmed people were shot. Here, most polling stations opened.

After oligarch Rinat Akhmetov, Ukraine's richest man and an important political broker, came out in the past fortnight against the separatist movement, he also ordered his factories to provide unarmed worker patrols to ensure order in the city. The separatist barricades have been removed, and workers were standing guard at polling stations. Voting proceeded smoothly but the atmosphere was tense.

"The ballots were delivered in the dead of night, and we weren't told they were coming until the last minute," said Sergei Pashkovsky, the head of