Counting had continued through the night after a strong turnout in Zimbabwe's first election without Mugabe, who was ousted by the military in November after 37 years in office.
Mugabe, wearing a dark suit and red tie, was greeted with cheers after casting his ballot but did not answer journalists' questions about who he voted for.
Mugabe, forced out under military pressure, had declared he would not vote for the ruling party he long controlled and called Chamisa the only viable candidate.
The Zimbabwe Electoral Commission has yet to release even preliminary results, but it has five days to release the final tally, and it says it is confident of doing so within that time frame.
"The results show beyond reasonable doubt that we have won the election and that the next president of Zimbabwe is Nelson Chamisa", said the Movement for Democratic Change's (MDC) Biti, who was the respected finance minister in the 2009 - 13 power-sharing government.
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.
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"Overall (there was) a huge amount of voting - especially young people, mostly in a very good atmosphere, generally peaceful, which is positive", he added.
The UN High Commissioner for Human Rights warned of intimidation and threats of violence in the run-up to polling day, but said it was encouraged to see open rallies and peaceful demonstrations. It suggested there was a deliberate effort to delay announcing the results, reflecting deep suspicion about the panel presiding over the election.
The commission said it would delay releasing any results of the presidential race until all the votes are collated. Mnangagwa, the former vice president, was sworn in as Mugabe's successor. The historic election is the first since the fall of longtime leader Robert Mugabe.
During Mugabe's rule, national elections were overshadowed by repeated voting procedure violations and the suppression of the opposition.
Elections are also taking place for 210 members of parliament and more than 9,000 councillors.