However if neither party wins over 50% of the vote, a runoff between Chamisa and Mnangagwa must be held on September 8, which would not include the other 21 candidates on the presidential ballot.
Long lines formed outside many polling stations in Harare, the capital, and elsewhere.
Chamisa was reported to the police by the electoral commission on Sunday, who said a press conference he had held on that day violated the electoral act which forbids campaigning on the day before the vote.
Elections observers from the European Union and the United States have also been allowed to monitor the vote - the first time since 2002.
"I can not vote for those who have tormented me", Mugabe said in a reference to Mnangagwa, who took office with the military's support. "He is young and can understand our plight as youth", said Ndumiso Nyoni, 20, a worker at a lodge in Lupane, western Zimbabwe.
With results expected to start trickling from Monday night, the focus now shifts to whether the vote was credible - a key pillar needed to rebuild the southern African nation's worldwide reputation and economy that was laid to waste under Mugabe's 37-year rule.
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"There are shortcomings that we have to check".
The bloc will deliver a preliminary report on the conduct of the election on Wednesday, as will the Southern African Development Community (SADC) and the African Union teams.
Despite Mugabe's troubled legacy, dozens of cheering Zimbabweans gathered outside the polling station in the capital where he voted.
"Overall (there was) a huge amount of voting - especially young people, mostly in a very good atmosphere, generally peaceful, which is positive", he added.
More than 5.5 million people were registered to vote and turnout was about 70 percent.
He emerged on the eve of the election to announce he would vote for the opposition, surprising Mr Mnangagwa who accused him of striking a deal with Mr Chamisa.
Piercing whistles and cheers greeted Chamisa as he voted outside Harare, who declared on Twitter that "minus the ballot mischief".
"This country is enjoying democratic space which has never been experienced before", Mnangagwa said after casting his vote.
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Mnangagwa, 75, has promised jobs and is regarded as being open to economic reform.
"Zimbabwe experienced a attractive expression of freedom and democracy".
In contrast, Mr Mnangagwa, thought to be 75, voted in a quiet polling station at a school in the central city of Kwekwe. They need to manually count the ballots cast and also have them verified.
The official result has to be announced within five days but there will likely be an indication of the outcome on Tuesday.
Nelson Chamisa's MDC Alliance has been warned "to safeguard victory" after the party announced victory before official confirmation from ZEC.
On Monday, he alleged there was a "deliberate attempt to suppress" voting in urban areas - MDC strongholds.
Although the campaign has been free of the systematic violence that marred previous polls, the MDC has repeatedly claimed it has been hindered by a flawed electoral roll, ballot paper malpractice, voter intimidation, bias in the electoral commission and handouts to voters from the ruling party.
The new government must tackle mass unemployment and an economy shattered by the seizure of white-owned farms under Mugabe, the collapse of agriculture, hyperinflation and an investment exodus.
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