Zimbabwe's president has urging political leaders to advocate for peace "as this day that ended in tragedy comes to a close".
Mnangagwa, a former deputy president who fell out with Mugabe and then took over from him, has said his showing in the election was "extremely positive" while urging people to wait for official results.
Worldwide observers who had initially given high praise for Zimbabwe's peaceful elections, issued more cautiously worded preliminary post-election results, alluding to some inconsistencies that could legitimize the opposition and its supporters' concerns.
Police spokeswoman Charity Charamba told state broadcaster Zimbabwe Broadcasting Corporation (ZBC) that the three people killed in the clashes had yet to be identified.
The government warns candidates they face prosecution and jail for prematurely announcing results.
A Zimbabwean soldier beats a man on a street in Harare on August 1, 2018 as protests erupted over alleged fraud in the country's election.
The European Union (EU) mission has criticised the delay in announcing the presidential results.
Gunfire crackled as troops, backed by armoured vehicles and a military helicopter and some with their faces masked, cleared the streets of opposition protesters.
"It matters little whether this heavy-handed response came on Mnangagwa's orders: evidence that the president lacks the authority to control the security forces will be just as damning in terms of the impact on Zimbabwe's global rehabilitation".
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Officers responded with tear gas and water cannon.
The ruling ZANU-PF party won a majority of seats in Parliament, the Election Commission said.
Chamisa said on Twitter he had won the "popular vote" in Monday's election, in which he challenged Mugabe's successor, Mnangagwa from the ruling ZANU-PF party.
Monday's polls had been meant to turn the page on years of violence-marred elections and brutal repression of dissent after Mugabe's 37-year rule was ended by a brief military takeover in November.
The government blamed the MDC opposition party for inciting the unrest and vowed to enforce a security crackdown, and President Emmerson Mnangagwa said he wanted an independent investigation into the killings and that he is seeking settle differences "peacefully". Why are the army here beating us? shooting us?
The opposition leader's spokesman Nkululeko Sibanda said the army's reaction in quelling disturbances in the capital was disproportionate and unjustified.
Support for Zanu-PF historically has been deepest in rural areas, particularly the party's Mashonaland heartland, where more than two-thirds of Zimbabwe's 17 million people live. He accused the commission of releasing the parliamentary results first to prepare Zimbabweans for a Mnangagwa victory. The presidential result has yet to be declared.
"We are exhausted of them stealing our votes".
A presidential candidate needs more than 50% of the vote to win outright. "There is arguably a strong desire to help Zimbabwe pave its way out the economic disaster of the last decades, but also an understandable suspicion of the real intent of the political and economic elite to change their ways".