Mnangagwa's conciliatory remarks came a day after soldiers occupied the streets of Harare, shooting live rounds and beating demonstrators, many of whom were throwing rocks and setting fires to protest alleged fraud in Monday's election.
"We call on the political leaders and the population as whole to exercise restraint and reject any form of violence while awaiting the resolution of the dispute and the announcement of the electoral results".
"The Zimbabwe Republic Police would like to confirm the unfortunate deaths of three people during the riots and melee which occurred in Harare Central business district", police spokeswoman Charity Charamba told state television.
Opposition protestors take to the streets of Harare, burning tyres and pulling down street signs.
"They are trying to protest so they can get fair results", said Elisha Pfigu, a 31-year-old street vendor who warily watched soldiers at an intersection.
The ruling ZANU-PF party won a majority of seats in Parliament, the Election Commission said.
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The commission is urging the public to remain calm and condemns the violence in the capital, Harare, on Wednesday that left three people dead.
Quiet descended as people waited for election results expected later in the day.
Worldwide observers from the Commonwealth and elsewhere are urging the release of the presidential results as soon as possible.
During the event, other independent Preliminary Statements were issued including by the African Union Election Observation Mission, Common Market for Eastern and Southern Africa (COMESA), SADC Parliamentary Forum (SADC-PF) and Electoral Commission Forum of SADC Countries (ECF-SADC).
The EU's Chief Observer, Elmar Brok, said he did not yet know if the shortcomings would have a material effect on the outcome of the vote, and he criticised the Zimbabwe Electoral Commission (ZEC) for being at times "one-sided".
The news was delivered by the Zimbabwe Electoral Commission (or ZEC) as they announced the results in dribs and drabs for the lower house of parliament.
European Union observers have listed several problems with the vote, including media bias, voter intimidation and mistrust in the electoral commission.
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The government has since pushed the blame on the MDC Alliance while the opposition party accuses ZEC and ZanuPF of conniving to rig elections which it believes it won convincingly.
MDC leader Chamisa has said outright that his own count shows he won the election, drawing government accusations of inciting violence.
What will happen in the presidential vote?
The situation in the country now is rather disappointing, according to many observers.
CIVIC society groups based in Matabeleland have blamed the two warring MDC-T factions, led by Nelson Chamisa (left) and Thokozani Khupe (right), for selling out their parliamentary seats to the ruling Zanu PF party by failing to resolve their differences ahead of Monday's general elections. "We won the popular vote and will defend it!"
Incumbent leader Mr Mnangagwa, 75, had promised a free and fair vote after a military coup overthrew his former boss, Mr Mugabe, ushering him into power past year.
Zimbabwe soldiers clash with opposition supporters as ruling party wins majority
Ballots are still being counted in the presidential election with results expected to be announced later this week. Riot police using water cannon and tear gas arrived to a chaotic scene of burning tyres and an unrelenting crowd.