The interior ministry says hundreds of electoral violations have been reported, but foreign observers say the vote appeared to be mainly smooth.
A total of 39 candidates were on the ballot paper, and with none receiving 50% the top two will go forward to the run-off on 21 April.
The Ukrainian president has significant powers over security, defence and foreign policy and the ex-Soviet republic's system is described as semi-presidential.
Who is Volodymyr Zelenskiy?
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Media captionThe comedian who could be president
Mr Zelenskiy is aiming to turn his satirical TV show Servant of the People - in which he portrays an ordinary citizen who becomes president after fighting corruption - into reality.
He has torn up the rulebook for election campaigning, staging no rallies and few interviews, and appears to have no strong political views apart from a wish to be new and different.
His extensive use of social media appealed to younger voters.
Mr Poroshenko, a chocolate magnate and one of Ukraine's wealthiest people, was elected in a snap vote after former pro-Russian President Viktor Yanukovych was toppled in the February 2014 Maidan Revolution, which was followed by Russia's annexation of Crimea and a Russian-backed insurgency in the east.
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Media captionUkraine's presidential elections: five things to know
The next president will inherit a deadlocked conflict between Ukrainian troops and the eastern separatists, while Ukraine strives to fulfil EU requirements for closer economic ties.
The EU says that about 12% of Ukraine's 44 million people are disenfranchised, largely those who live in Russia and in Crimea, which Russia annexed in March 2014.
Mr Poroshenko aimed to appeal to conservative Ukrainians through his slogan "Army, Language, Faith".
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Petro Poroshenko voted with his wife
He says his backing for the military has helped keep the separatists in check. He also negotiated an Association Agreement with the EU, including visa-free travel for Ukrainians. During his tenure the Ukrainian Orthodox Church has become independent of Russian control.
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Yulia Tymoshenko stood for president twice before
Yulia Tymoshenko served as prime minister and ran for president in 2010 and 2014. She played a leading role in the 2004 Orange Revolution, Ukraine's first big push to ally itself with the EU.