Some 1.8 million voters were registered to decide the successor to incumbent Peter Obi, who steps down in March next year after completing a maximum two, four-year terms.
Fears of violence linked to the yesterday's election saw stringent security measures put in place in the mineral-rich southeast state, with restrictions on movement and a heavy police presence.
The state capital Awka stayed calm throughout the day, with security forces mounting road blocks and patrols, an AFP reporter in the city said.
Earlier in the day Idowu had said the election was "going fine for now but not without isolated challenges" such as the delivery of election material to some polling stations.
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In the town of Obosi, voting was postponed due to delivery problems and will now take place today.
The electoral commission said results were expected in 24 to 48 hours.
In all, 23 candidates are vying for the governorship, including Nigeria's main opposition All Progressives Congress (APC), which has yet to hold Nigeria's presidency since the country returned to civilian rule in 1999.
Jonathan, a southern Christian, is tipped to run again but has faced dissent within his party from some who believe he should make way for a northern, Muslim candidate.
Vote-rigging and election-linked violence have often marred recent polls in Nigeria. In 2011, some 800 people died in the Muslim-majority north after Jonathan's election.
The last election in Anambra in 2010 was judged "credible" by US observers, although they criticised poor organisation at some polling stations.