Buhari, a former military ruler who was later elected president, is seeking a second term in charge of Africa's most populous nation and top crude producer.
The army also confirmed a "futile" attack on a security outpost in Geidam in Yobe state.
Shortly before polls opened, a series of blasts rocked the northeastern city of Maiduguri, which has been repeatedly hit by Boko Haram Islamists.
In a statement issued on the day of the postponement, the All Progressives Congress (APC) said the People's Democratic Party (PDP) wanted to halt the momentum of its candidate, President Muhammadu Buhari.
The electoral commission now swears it is ready.
" I have been voting since 1999".
Nigeria's president says he will be congratulating himself at the end of the election after he was among the first Nigerians to cast their ballots.
Observers said the delay of the election from last week, blamed on logistical challenges, could favour President Buhari and the ruling party, with some Nigerians saying they didn't have the resources to travel to their place of registration a second time. He previously pledged to accept the results, provided they are credible. He said polls opened two hours late and machines to read voting cards were not working.
Buhari called the voting process smooth, but in other parts of the country some officials reported concerns that some polling stations were late to open and that the heavy security presence could intimidate some potential voters.
The wife of the president looks to have allowed him so to do with the entire episode drawing laughter among the people present at the booth.
Voting began from 0700 GMT for the presidential and parliamentary elections across the country.
Police patrol vehicles were seen moving around Borno's capital, Maiduguri, on Friday.
Buhari called the voting process smooth, but some polling stations in other parts of the country were late to open and some officials anxious that heavy security could intimidate potential voters.
Security sources said militants had struck at parts of Maiduguri and a Reuters witness said he had heard gunshots and Nigerian air force jets were flying overhead.
But the Boko Haram extremist group, its Islamic State-affiliated offshoot and various agitators across the country have other plans.