The UN and former colonial power the UK both expressed concern about the violence, in which three people were killed after troops opened fire.
Parliamentary results gave victory to the ruling Zanu-PF party in the first vote since the ouster of former ruler Robert Mugabe.
But the opposition says Zanu-PF rigged the election.
The result of the presidential election has yet to be declared. The MDC opposition alliance insist their candidate, Nelson Chamisa, beat the incumbent President Emmerson Mnangagwa.
UN Secretary General Antonio Guterres urged Zimbabwe's politicians to exercise restraint, while UK foreign office minister Harriett Baldwin said she was "deeply concerned" by the violence.
The US embassy in Harare urged the army to "use restraint" on Twitter, saying the country had an "historic opportunity" for a brighter future.
Human rights group Amnesty International meanwhile called on the government to open a probe into the army's actions.
Amnesty's acting secretary general Colm O Cuanachain said in a press release that the "militarisation" of the election aftermath "is muzzling freedom of expression, association and assembly".
"People must be guaranteed their right to protest," he said.
What happened after the vote?
The country's electoral commission said that Zanu-PF won a two-thirds majority in the national parliament - prompting protests in the capital, Harare.
The country's Home Affairs Minister Obert Mpofu said the government "will not tolerate" the protests.
The opposition "are testing our resolve", he said, "and I think they are making a big mistake."
A spokesman for opposition candidate 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 today."
The Zimbabwe Electoral Commission (Zec) has so far announced victory in 140 seats for Zanu-PF, with 58 for the MDC Alliance, ZBC state media reported. There are 210 seats in the National Assembly's lower house.
More than five million people were registered to vote, and there was a high turnout of 70%.
ZBC had reported that the electoral commission would announce the presidential results at 12:30 local time (10:30 GMT) on Wednesday, but only parliamentary results were read out.
The BBC's Shingai Nyoka reports that the announcement on the presidential poll was not made because representatives of some of the 23 candidates had failed to turn up to verify the results.
A presidential candidate needs more than 50% of the vote to win outright. Otherwise, a run-off election will be held on 8 September.
What are election observers saying?
The EU mission criticised the delay in announcing the presidential results. Zec has until Saturday to do so.
It said it had observed several problems, including media bias, voter intimidation and mistrust in the electoral commission, adding that there was an "improved political climate, but un-level playing field and lack of trust".
This is the first time in 16 years that the government has allowed EU and US election monitors into the country.
The African Union mission has said the elections "took place in a very peaceful environment" and "were highly competitive".
It added that it could not confirm opposition parties' complaints of vote-buying, intimidation by the state and bias by traditional leaders.
A preliminary report by the Southern African Development Community (Sadc) observers said the elections were largely peaceful and conducted in accordance with the law.