Polling places reported long lines of voters waiting to cast ballots but some opened late and others had missing materials.
Calm was reported across the country, although police reported Karaga's People's National Convention constituency chairman, Alhassan Ze, died in an accident en route to vote, Myjoyonline reported.
Nearly 14 million registered voters are expected to elect the next president. Also up for a vote are 275 seats in Parliament sought by 1,300 candidates, Myjoyonline said.
If neither Mahama, of the National Democratic Party, nor Akufo-Addo, of the New Patriotic Party, wins more than 50 percent of the vote, a runoff will be conducted Dec. 28.
A parliamentary candidate needs a simple majority.
Mahama took over as president after John Atta Mills died suddenly in July.
For the first time, a biometric voters' register with electronic verification is being used in the election, officials said.