Analysts say the vote is too close to call, with the outcome set to hinge on which man voters most trust to revamp an economy still struggling to recover from a 2016 recession.
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. Atiku, a former vice president, has pledged to expand the role of the private sector, Presstv Reported.
The two candidates lead a field of more than 70 challengers.
Last Saturday, the election was postponed around five hours before polling stations were due to open and there are concerns that the week's delay might hurt turnout.
The electoral commission blamed logistical factors for last week's delay and denied political pressure was behind the decision. Presidential elections in 2011 and 2015 were also delayed over logistics and security concerns.
Buhari, who voted in his hometown of Daura in the northern state of Katsina, said "I will congratulate myself" when asked by reporters if he would congratulate his rival, should his rival win the election.
Atiku later cast his ballot in the eastern Adamawa state.
"I look forward to a successful transition," he told reporters shortly after voting.
Nigerians queued at polling booths around the country where voting was set to begin at 8:00 a.m. local time (0700 GMT).
Several polling stations across the country were, however, slow to open, Reuters witnesses said. The country has 72.8 million eligible voters.
"I’ve been to 10 polling units today. I’ve been redirected many times," said Victor Kanoba a voter in Lagos.
"This was somewhat expected given the logistical challenges of getting all of the materials to the polls in time for opening at 8 am," John Tomaszewski, an observer with the joint US National Democratic Institute and International Republican Institute delegation, said referring to the delays.
"This will be something to watch throughout the day."
In the capital Abuja, Chukwunwike Ogbuani, a lawyer, said he was worried by the delay in opening polling stations.
"This polling booth they say has about 20,000 registered voters... if there is at least up to 50 to 60 percent turnout it will be difficult to finish in a day. I don’t see everybody that is here voting within the stipulated time."