With all 10 provinces declared, Mr Mnangagwa won 50.8% of votes to 44.3% for opposition leader Nelson Chamisa.
Police removed opposition officials from the electoral commission stage when they rejected the results.
The chairman of Mr Chamisa's MDC Alliance said the count could not be verified.
By narrowly winning more than 50% of the vote, Mr Mnangagwa avoids a second run-off election against Mr Chamisa.
The president said he was "humbled" on Twitter, and called the result "a new beginning".
Mr Chamisa has insisted he has won the presidential poll, telling reporters earlier on Thursday the ruling Zanu-PF party was "trying to bastardise the result", something that "we will not allow".
But the Zimbabwe Electoral Commission (Zec) said there was "absolutely no skulduggery".
Mr Mnangagwa's supporters celebrated after the results were announced
Opposition supporters protested in Harare over alleged vote-rigging, which led to six deaths on Wednesday.
The elections were the first since long-time ruler Robert Mugabe, 94, was ousted in November last year.
The polls were intended to set Zimbabwe on a new path following Mr Mugabe's repressive rule.
Harare was a ghost town on Thursday following Wednesday's violence, as troops patrolled the city centre ordering people to "behave".
The government has blamed the opposition for the violence
Mr Mnangagwa said the government was in talks with Mr Chamisa to defuse the crisis and proposed an independent investigation to bring those who were behind the violence to justice.
"This land is home to all of us, and we will sink or swim together," Mr Mnangagwa said in a series of tweets.
What happened after the vote?
The day after the election, the MDC Alliance said Mr Chamisa had won the presidential election, pre-empting an official announcement and prompting its supporters to celebrate in some areas of Harare
When Zec announced that Zanu-PF had won the parliamentary vote by a landslide on Wednesday, things turned nasty.
The opposition supporters were are also angered by the delay in announcing the presidential results
Home Affairs Minister Obert Mpofu said the government would not tolerate such protests.
The opposition "are testing our resolve, and I think they are making a big mistake", he said.
A spokesman for Mr Chamisa condemned the deployment of soldiers and the subsequent loss of life.
"Soldiers are trained to kill during war. Are civilians enemies of the state?" he asked.
"There is no explanation whatsoever for the brutality that we saw."
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What are the full results?
Zec announced the results from the tenth and final province, Mashonaland West, late on Thursday after days of waiting.
The European Union and Commonwealth missions earlier criticised the delay in announcing the presidential results.
This is the first time in 16 years that the government has allowed EU, Commonwealth and US election monitors into the country.
Earlier in the week, Zec announced the parliamentary results, giving Zanu-PF 144 seats, the MDC Alliance, which is made up of seven parties, 64 seats, and the National Patriotic Front, formed by Mugabe loyalists after he was ousted, one.
Although Zanu-PF won by a landslide, its majority has shrunk since the 2013 election when it obtained 160 seats and the MDC, then led by the late Morgan Tsvangirai, 49.
More than five million people were registered to vote in Monday's poll, and there was a turnout of 70%.
The electoral commission says 70% of registered voters took part in the election