Mnangagwa, a former deputy president who fell out with Mugabe and then took over from him, has said his showing in the presidential polls was "extremely positive" while urging people to wait for official results.
Electoral Commission Priscilla Chigumba told reporters that "the atmosphere has remained peaceful" and the commission has not received any major complaints about how the election was conducted.
Winning resoundingly... We now have results from the majority of the over 10 000 polling stations.
"I am not shy to say I voted for Chamisa".
Mnangagwa also called on "citizens and candidates alike to exercise responsibility and restraint by patiently waiting" for official results.
"Vote counting is underway and we should be able to release the first results around 3pm".
The Zimbabwe Electoral Commission said the first results were expected Tuesday afternoon, with the final tally expected within five days.
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Zimbabwe appeared to be heading for a contested result in the historic election, raising the likelihood of a bitter battle over allegations of fraud and malpractice from both sides.
The election is a two-horse race between 75-year-old Mnangagwa, a long-time Mugabe ally, and 40-year-old Nelson Chamisa, a lawyer and pastor who is vying to become Zimbabwe's youngest head of state.
Zimbabwe's main opposition is claiming that voting results forms were not posted at more than one-fifth of the country's polling stations as the law requires.
"We will not steal their choice of leaders, we will not subvert their will", Chigumba said.
Officials neared the end of vote counting a day after millions peacefully cast their ballots in a process closely watched by global monitors, who have yet to announce whether the election was free and fair.
A voter in Harare, the capital, said Zimbabwe is anxious to hear the election results as soon as possible.
As well as electing a president, Zimbabweans were voting for 210 members of parliament and more than 9,000 councillors.