It is seen as a test run of how next year's national presidential vote will be conducted.
Over 1.8 million voters are eligible to cast ballots at 4,623 polling stations scattered around one of the country's most densely populated states.
A total of 25 candidates are vying for the post of governor but no clear favourite has emerged among the four frontrunners in the race.
The candidates include Peter Obi, the current governor who represents the All Progressives Grand Alliance party.
"This election is already being hit about questions over its legality only two hours since the polls opened"
Yvonne Ndege, Al Jazeera's correspondent
Governors of the 36 Nigeria's states are powerful as they pick presidential candidates at party conventions.
Al Jazeera's Yvonne Ngege, reporting from Awka, said the election was already mired in controversy.
"This election is already being hit about questions over its legality only two hours since the polls opened," she said.
"One of the contestants in this competition is saying [that] Inec - the Independent National Electoral Commission - is not fit to conduct this election because president [Umaru] Yar'adua failed to sign in several commissioners before going on indefinite sick leave."
But our correspondent added that she had spoken to the head of the Inec who said "he hasn't received any court orders to stop the election".
"I've also spoken to the police who say they haven't received any court orders to stop the election and [that] it will go ahead no matter what," she said.
The election is being held amid growing uncertainty surrounding the president's illness, and threats of a renewed oil war after rebels in the Niger Delta abandoned last week a truce citing lack of progress in peace talks in the president's absence.
The governing People's Democratic Party (PDP) led by Yar'adua - now in a hospital for more than two months - controls all but eight of the 36 states.
Anambra is one of those that have been in opposition hands.
|The election is seen as a test run of how next year's national vote will be conducted [AFP]
But the PDP is going into the Anambra election divided after the party executive imposed Chukwuma Soludo, a former central bank governor, as its candidate, a move that led some party stalwarts to defect to the opposition.
Observers said the conduct of the election will "test ground for what is going to happen in the country".
Akuro George, leader of the Nigerian Bar Association (NBA) team of poll observers in Anambra, said: "This election is deemed to be very critical towards the electoral reforms that we have been clamouring for in this country.
"If we don't get it right, it will be an indication that we have not changed our mindset."
The last presidential vote which brought Yar'Adua to power was marred by widespread rigging and voter intimidation.
Yar'Adua vowed to improve the running of elections in Africa's most populous country after a court decided to keep him in office despite the irregularities.
Although the bill proposing electoral reforms has yet to be passed, analysts say Nigeria's problem is not much about the law, but its application.
"There is nothing wrong with the electoral laws ... but everything wrong with the average Nigerian politician and the electoral empire," Donald Awoor, a commentator, said.
Presidential elections are due in April next year and will see a second successive democratic handover between civilian rulers, after a long period of military rule ended a decade ago.