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 the vote outright without a second round. Former prime minister Yulia Tymoshenko was trailing far behind, with about 13%. Official results are expected overnight.

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 militia or simply for 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." He said that his first trip as president 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 repeated the oft-cited argument that they wanted Poroshenko to exceed the 50% threshold so the election would finish without a run-off 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 of Poroshenko. He voted for controversial radical politician Oleh Lyashko, the only candidate who had gone to the restive regions in the east where he has taken 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 hot spots." 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. "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 the separatist strongholds such as Slavyansk there would be no voting, but more surprisingly, in Donetsk, a city of close to 1 million 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, the majority of 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 the electoral committee at polling station 239, opposite the charred shell of the regional administration, set on fire during clashes this month. "It will be the same thing tonight. We don't yet know who will pick up the ballots, and where they will be taken, but we've been told it will be done under tight security. They will tell us the details at the last minute."

Across town at School Number Seven, there are usually two polling stations, but only one had opened. As a result, half of the people who came to vote were turned away as they were 'not on the list'. Roman Moroz, head of the electoral commission, said that 9 of its 12 members had pulled out over the past week, forcing him to drag his friends along to make up the numbers and ensure the polling station could open. The original members had been intimidated or received threats, he said.

In the capital Kiev, queues at polling stations stretched for an hour or more; at the few that had opened in Mariupol, the turnout at two different polling stations by 3pm was under 20%.

There is genuine anger in the east where, in the past few weeks, many people have become more convinced by separatist ideas. There were, however, many people who wanted to vote but were unable.

"Of course I would have voted if I could have," said Sergei, 29, who was walking along the riverbank in the city. "True, I'm not sure who for, as none of the candidates are very inspiring, but anyone is better than these idiots in masks playing at war. The city is sick of them. It is time to get back to normal."

However, with a proliferation of armed groups, increasing paramilitary activity, and a population that remains deeply sceptical of Kiev – even as many people tire of the separatists – regaining control will not be an easy task for the country's new president.

Poroshenko will also have to deal with an ongoing economic crisis, with the national currency, the hryvnia, continuing to fall and public debt at a huge level. The country received a bailout from the International Monetary Fund this year tied to painful social cuts and reforms. Poroshenko will also need to steer a delicate geopolitical path, moving the country towards closer ties with Europe demanded by the Euromaidan protests that swept out the government of Viktor Yanukovych in February, while improving hostile relations with Russia, its often belligerent large neighbour.

Poroshenko has pledged to sign as soon as possible the economic part of an association agreement with the European Union, the political half of which was signed in March. The agreement will establish a free trade area and take steps toward visa-free travel, while committing Ukraine to economic and judicial reforms. He will also have to prove he can usher in a new type of politics, free of the corruption and mismanagement that dogged the Yanukovych regime.

He has said he will not seek to join Nato, a controversial idea that has split the population and worried Russian leaders. In a major sign that the Kremlin was softening its stance on Ukraine, President Vladimir Putin said on Saturday that Russia would work with the Kiev government after the presidential vote. Previously, Russia has refused to recognise the regime, arguing it came to power through an armed coup.