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The cabinet minister was supposed to appear on the radio at 8.10am, but at that time was stuck in an elevator in the BBC's Broadcasting House in central London which had stopped working.
Having been freed after half an hour, Mr Gove jokingly said staff had "successfully levelled me up" and released him.
Today programme presenter Nick Robinson told listeners that Mr Gove had been stuck in the lift "for some time" but was "keeping cheerful" and had offered to conduct the interview by phone from inside the lift.
"I wish I could say this is a joke, it is not a joke and it is not very funny for Mr Gove and the security man," Mr Robinson said.
Eventually making it onto the airwaves, the levelling up secretary said: "After more than half an hour in the lift, you successfully levelled me up, so I'm delighted to be here."
Mr Gove also joked that the ordeal could give "ammunition" for scriptwriters of W1A - a mockumentary sitcom television series that satirises the management of the BBC.
Informed that a "free Michael Gove" hashtag was circulating on social media, Mr Gove said: "I suspect there were rather more people hoping I would be incarcerated for longer."
Mr Gove had been touring broadcast studios to talk about the action he is taking to tackle the cladding scandal.
The levelling up secretary later said he had texted BBC chairman Richard Sharp in order to be freed from the lift.
Speaking on LBC, Mr Gove said: "I'm talking to you from BBC News Broadcasting House but in a scene, perhaps, I think more appropriate for The Thick Of It or W1A, I was trapped in a BBC lift for half an hour between 7.45 and 8.15.
"But, thanks to the good offices of the Today programme, and also thanks to my texting the BBC chairman, I was liberated."
He added: "He made sure that a crack engineering team were dispatched. As someone pointed out, even though we had to wait half an hour, eventually I was levelled up."
Mr Gove said it was "very good advice from both a health and a safety point of view" to take the stairs out of the building.
The BBC apologised after the incident, with a spokesman saying: "We're sorry Mr Gove was stuck in one of our lifts, but we're glad he was later able to take part in the interview."