Mondlane went into hiding about two weeks ago, after his aide and lawyer were killed as they were preparing to challenge the results.
Daniel Chapo, the candidate of the ruling party Frelimo, was officially declared the winner with over 71% of the vote, as opposed to Mondlane's 20%.
Mondlane didn't specify exactly where the alleged assassination attempt had taken place.
In the video, he said he had been staying in the affluent Sandton neighbourhood in Johannesburg.
"I was with my wife and my daughter, running from one place to another," he said.
South Africa's foreign ministry told the AFP news agency they didn't even know Mondlane was in the country and that the matter should have been reported to the police.
The BBC reached out to South African police, but they didn't respond.
Mondlane went into hiding before the results were announced, claiming his home had been surrounded by members of the security forces.
He has called on Mozambicans to hold nationwide demonstrations against the election results, which are being challenged in court by the Podemos party which backed Mondlane.
The protests have led to violent clashes with the police and several people have been killed, with dozens injured.
The internet and social media have also been restricted.
A general strike called by Mondlane has continued, despite the prime minister's call for people to return to work.
Three children were reportedly shot dead by police in different parts of the capital Maputo on Monday, while eight people were reportedly killed in the northern province of Nampula over the weekend.
Two different Frelimo offices were reportedly attacked in Nampula, prompting the police to respond with tear gas, rubber bullets, and live ammunition, killing one person.
The demonstrators then set fire to the house allegedly belonging to a police officer accused of killing a protester.
Mondlane has been rallying people to come together for a "final march" in Maputo, on Thursday.
It is due to be the end of a seven-day protest he called last week.
Source: BBC
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