After a surprise last-minute election delay a week ago, officials believed another postponement would be disastrous, even if it meant a bumpy day of voting.
In southern Rivers state, unknown gunmen shot dead a prominent local member of Buhari's All Progressives Congress party and his brother, police said.
The statement said the attack was repulsed and there were no casualties, adding that the situation is now calm in Geidam, allowing people to vote.
Boko Haram, the Islamic extremist group, is active in northern Nigeria.
A civilian militia source in the Borno state capital, Maiduguri, told AFP "at least 13" blasts were heard throughout the city at about 6:00 am (0500 GMT).
"With regards to the challenges associated with Smart Card Readers in some polling units, most of these challenges were resolved and were resolved on time".
The commission also vowed to prosecute any of its staff who failed to use the Smart Card Readers for purposes of accreditation, saying where the card readers do not work up till 6pm on election day, the standard procedure is to shift the exercise to the next day. Delays were reported in Delta, Anambra and Akwa Ibom states as well as in Lagos, Nigeria's largest city.
According to the Independent National Electoral Commission, INEC, elections are expected to begin at 8 am.
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President Buhari was one of the first in the queue when the almost 120,000 polling stations began opening at 0700 GMT.
Some Nigerians are reporting low turnout or the late opening of polls more than one hour into the presidential election. Ibrahim Mustapha, one Yola voter, was annoyed.
In a televised morning address on the eve of the vote, Buhari asked Nigerians to "cast aside doubt and have faith that INEC will rise to the occasion" on Saturday. But would-be voter Grace could not find her number on the list at a polling station in Abuja.
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.
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. Nigerians began voting for a new president on 23 February after a week-long delay that has raised political tempers, sparked conspiracy claims and stoked fears of violence. Four years ago, Buhari was the first-ever opposition candidate to defeat an incumbent president. Abubakar points to his business success in making sweeping pledges to turn the economy around but is dogged by corruption allegations.
"The government made the decision - in consultation with the security agencies - that it is better and more useful for the security agents, including soldiers, police, Civil Defence, DSS, etc to focus their attention on providing security for the election throughout the state".
Security is also a major concern; in the northeast, the Boko Haram insurgency continues, and in central Nigeria, herder-farmer conflicts over land and grazing rights have cost hundreds of lives.