"Awaiting ZEC to perform their constitutional duty to officially announce the people's election results and we are ready to form the next government", said Chamisa, who is vying to become Zimbabwe's youngest head of state, on his official Twitter feed.
Millions of people voted peacefully on Monday in the first poll since the army removed Robert Mugabe from power a year ago.
Mugabe on Sunday called Chamisa the only viable candidate and rejected Mnangagwa and the ruling party, saying that "I can not vote for those who have tormented me".
Liberia's former President Ellen Johnson-Sirleaf, who is monitoring the poll on behalf of the US-based National Democratic Institute, told the BBC that the long queues showed that Zimbabweans were enthusiastic about voting, without any kind of repression.
"While investors remain sceptical over whether Mugabe's former right-hand man has indeed turned over a new leaf, Mnangagwa's charm offensive with Western governments and businesses has at least given him a credible lifeline at the poll", said Verisk Maplecrodt analyst Charles Laurie in a note.
Zanu-PF, however still have wide support across the country with Mugabe's former party representing the end of colonial ties and white-minority rule.
"I can not vote for those who tormented me".
Deposed president Mugabe, who ruled Zimbabwe for 37 years, voted in Highfield, Harare.More news: Mourinho mocks ref after Man Utd humbling: He's come from baseball!
He did not‚ however‚ agree with Chamisa's view that the Zimbabwe Electoral Commission was biased towards the governing Zanu-PF party.
In another message to the people of Zimbabwe, President Mnangagwa said it is now clear that his rival, Nelson Chamisa has struck a deal with former president Robert Mugabe and it is up to Zimbabweans to vote for Mugabe under the guise of Chamisa or vote for a new Zimbabwe under his leadership. "I hope the choice of voting tomorrow will thrust away the military government and bring us back to constitutionality", Mr Mugabe said. "He is a citizen".
Chamisa, meanwhile, said at a news conference that he welcomed the vote of Mugabe or any other Zimbabwean and that "you don't discriminate against voters".
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 EU has not yet made a conclusion on how to judge the vote, he said.
A recent Afrobarometer survey of 2,400 people put Mnangagwa on 40 per cent and Chamisa on 37 per cent, with 20 per cent undecided. "We are leaving no stone unturned.to make sure we win this election".
In an astonishing public intervention on Sunday, Mugabe said he would not vote for the party he founded, which removed him from power last November, and would instead be casting his vote for the MDC.
With 5.6 million registered voters, full results of the presidential, parliamentary and local elections are due by August 4.
If no presidential candidate wins 50 per cent of the vote, a runoff will be held on September 8.