The 47-year-old pop star broke with tradition by handing over the selection of one of her eight allotted tracks to the British actor.
He surprised her with a reading of As I Look Up, written by his father, the late poet Dominic Sasse.
“This is a first for us, I have to say, in our 74-year history,” said presenter Kirsty Young.
The work includes the lines: “We would smear our mouths with ardent kisses and cry aloud from loving, as we lay beneath the arches of that careless wanton day.”
At the end of the reading, a clearly emotional Minogue said: “I’m a mess.”
She added: “I think I stopped breathing for half of that. Breathe, Minogue. That was absolutely beautiful. I have tears of joy. Tears of joy, right now.”
The couple met in September on the set of Sasse’s US television series, Galavant.
“I have a love in my life, which is just a beautiful thing and I’m just kind of on cloud nine most of the time because of Mr Joshua Sasse, my beau,” she told BBC Radio 4’s Young.
“Who knows if family’s on any one of my horizons?” the Australian stated. “I don’t know. But if that was to happen, that would be incredible.”
Minogue also discussed her 1989-1991 liaison with tragic INXS rocker Michael Hutchence, who died in 1997.
“I met a new person and I fell in love. Yes, he was this wild guy, he was the rocker, he was all of that. But I’m always at pains to say, he was Byron-esque, he was poetic, he was cultured and hilarious and tender, he was all of these other things.
“I guess, I was at the perfect age, I was 21 years old, to get the butterfly wings and go out into the world and we collided at that time and I guess he just fast-tracked some of it. Anyway, it was a glorious time. I loved it.”
The relationship raised eyebrows due to the Neighbours star’s girl-next-door image. She went on to relate how supportive he was of her career.
“He was uber, uber famous, he’s an icon already and he was in the middle of the audience just gazing up at me and willing me with all of his heart and his love to do well. He believed in me and it was a time when so many people didn’t. It’s getting me... “ she said as her voice broke with emotion.
Need You Tonight by INXS was also selected for the island. “I performed it on my last world tour,” she explained. “I literally would have a word to him in my mind before I performed it every night.”
Young brought up Minogue’s breast cancer diagnosis at the age of 36.
“I thought, I’m not done yet. I’m not finished. This is not where I want to say goodbye. Of course you learn a lot about yourself and about other people and about life and what resilience means and what love means and what good health means,” she stated.
The pint-sized star left Australian soap Neighbours for a pop career, which was ignited by producers Stock, Aitken & Waterman. But the media attention was unwelcome at times.
“There was definitely a period where I had my version of a breakdown,” she admitted. “It’s probably a really cheesy thing to say, but Kylie does mean boomerang so I do my best to come back.”
In a lighter moment, the award-winning star opened up about her obsession with Prince and selected Purple Rain for the island.