Western observers barred
Because of the high turnout election officials granted long extensions to the opening hours of some polling stations.
To be declared a winner, a presidential candidate must win more than 50% of the vote. If no candidate reaches this mark, a run-off will be held on 11 September.
The elections were the first to be held under the new constitution approved in a referendum in March this year.
The government barred Western observers from monitoring Wednesday's elections, but the African Union (AU) and the Southern African Development Community (SADC), as well as local organisations, have been accredited.
The AU described voting at most polling stations as "orderly and peaceful", while the main domestic monitoring agency, the Zimbabwe Election Support Network, said aside from the long queues voting was "smooth".
Former Nigerian President Olusegun Obasanjo, who heads the AU monitors, told Reuters there had been no "serious incidents that... would not reflect the will of the people".
At a news conference as polls closed, police spokeswoman Charity Charamba warned that "all people who may wish to announce the results of elections before the ZEC [Zimbabwe Electoral Commission] does... risk being arrested".
The warning comes after monitoring groups, newspapers and social media users reportedly planned to publish provisional tallies.
There have been numerous complaints that voters were unable to find their names on the electoral roll.
According to villagers, MDC polling agents and local election observers, irregularities were recorded in parts of rural Masvingo district.