The election follows decades of repressive rule which has brought severe economic challenges to Zimbabwe.
The results of the presidential elections are due by August 4.
The elections were the country's first since Robert Mugabe stepped down last year, ending a 37-year presidential reign.
Tomorrow's vote is the first since Mr Mugabe was forced to step down in November after a de facto coup and, as such, is a major national test.
Queues of eager voters snaked around the streets of the capital Harare from before sunrise.
The first group of voters were mainly elderly people but there were some youngsters too. Long lines of voters were waiting outside some polling stations. Zimbabwe's future is in the balance and nothing is certain.
"We have seen that liberation movements never make good government".
"By the end of the day today we should be very clear as to an emphatic voice for change, the new, and the young - I represent that", Chamisa said as he voted in Harare, surrounded by vocal supporters.
There were no immediate reports of violence, said Andrew Makoni, chairman of the Zimbabwe Election Support Network, an association of 34 civil rights and religious organizations.
Elections under Mugabe were often marred by intimidation, rigging and violence but the consensus is the build-up to this vote has been better than before, although Chamisa complained about a flawed voters' roll and opaque ballot paper printing.
Whoever wins, we wish him well ...
He did not‚ however‚ agree with Chamisa's view that the Zimbabwe Electoral Commission was biased towards the governing Zanu-PF party.
He lingered in his cardboard voting booth, his daughter at his elbow, and placed his three ballots carefully in their boxes.
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Mr Mnangagwa tweeted that Zimbabweans should vote with "peace in [their] hearts".
"We will sink or swim together", he added.
Reuters spoke to several global observers who said the voting process had been slow at some stations but it did not appear to them to be intentional.
Average turnout was 75 per cent, higher than expected.
The President had time to chat with the polling officers, inquiring on how many people were allocated to vote at the polling station.
Hundreds of worldwide observers have been deployed to ensure the vote goes smoothly, but the opposition has repeatedly alleged irregularities in the voter roll.
Important elections have been taking place in Zimbabwe in southern Africa.
Zec head Priscilla Chigumba said 90% of polling stations had opened on time.
Voting closed at 7 p.m. (1700 GMT).
He backed Chamisa to rebuild a capable state in Zimbabwe and revive the economy and said the SAPDC would continue to partner with him to rebuild the country.
What are foreign observers saying?
European Union (EU) chief observer Elmar Brok said "transparent, credible and inclusive elections are crucial for Zimbabwe".
23 Zimbabweans are vying for the top seat but research organisation, Afro Barometer conducted a survey recently and put Mr Mnangagwa in the lead with 40% followed by Mr Chamisa with 37%. We have not found out whether there is coincidence or bad organisation.
Mr Mugabe has been a controversial figure though.