The Hollywood star said Jasmine, four, and two-year-old Tiana experienced "a little sore throat" but no other symptoms, however he and wife Lauren Hashian had a "rough go".
Despite being "very disciplined" and being in lockdown since March, Johnson said he ended up catching the "relentless and unforgiving" virus from "very close family friends".
Johnson, 48, announced the news in an Instagram video, confirming he has recovered from the illness while urging fans to take the threat seriously.
He said: "I can tell you that this has been one of the most challenging and difficult things we have ever had to endure as a family.
"And for me personally too, as well. And I've gone through some doozies in the past... but testing positive for Covid-19 is much different than overcoming nasty injuries, or being evicted, or even being broke, which I have been more than a few times."
The Fast & Furious star added: "But I am happy to tell you guys that we as a family are good.
‘We are on the other end of it, we are on the other side. We are no longer contagious and we are, thank God, we are healthy."
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Johnson advised fans to live a healthy lifestyle in order to maximise the body's defences against illness and urged people to wear a mask.
He said: "It baffles me that some people out there, including some politicians, will take this idea of wearing masks and make it part of a political agenda."
Johnson added: "It has nothing to do with politics. Wear your mask. It is a fact and it is the right thing to do and it is the responsible thing to do."
He is not the only celebrity to publicly announce a coronavirus diagnosis.
Tom Hanks and his wife Rita Wilson were among the first to do so, while others including Idris Elba, Bryan Cranston and Antonio Banderas have also fought the virus.