However, many analysts have said they expect a win for Mnangagwa, a Zanu-PF veteran who replaced long-time president Robert Mugabe after he was ousted in a military coup in 2017.
The Zimbabwe Electoral Commission (ZEC) said of 210 parliamentary seats, 153 had been counted with Zanu-PF winning 110 and the MDC Alliance 41.
Ninety-four-year-old Mugabe, who was accused of political violence and rigging elections during his almost four decades in power, voted in Harare alongside his wife Grace.
Anti-riot police dispersed a few hundred opposition protesters who were attempting to enter the Zimbabwe Electoral Commission's offices.
The election commission has said there has been no cheating and it needed time to collate the votes.
Both the opposition and Western election observers have urged that presidential election results be released as soon as possible.
Even before the violence, European Union observers questioned the conduct of the presidential and parliamentary poll, the first since Mugabe's forced resignation after almost 40 years in charge of the Southern African nation.
More than five million people were registered to vote, and there was a high turnout of 70%.
Zimbabwe's ruling party has won a majority of seats in Parliament, the electoral commission announced Wednesday, as the country braced for the first official results of the presidential election.
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But the MDC earlier accused the election commission of deliberately delaying results in favour of the ruling party.
He tweeted: "We won the popular vote & will defend it!" They believe the election has been stolen, and are demanding the MDC be announced as victor.
A presidential candidate needs more than 50% of the vote to win outright. "Equally, we hold the party and its leadership responsible for any loss of life, injury or damage of property that arise from these acts of political violence which they have aided and abetted", he said.
Mnangagwa was allegedly involved in violence and intimidation during the 2008 elections when then opposition leader Morgan Tsvangirai pulled out of the run-off after attacks claimed the lives of at least 200 of his supporters.
This is the first time in 16 years that the government has allowed European Union and United States election monitors into the country.
Zimbabwe's President Emmerson Mnangagwa called for calm on Wednesday as opposition supporters clashed with police in central Harare ahead of the announcement of this week's presidential election results.
Meanwhile, EU observers have said the election was largely peaceful, but the use of state resources, coercion and bias by "traditional leaders" meant a level playing field was not "truly achievable".
"The atmosphere has remained peaceful" and the commission has not received any major complaints about the election, she said.
"There is no way that ZANU-PF will accept an MDC victory".