The 82-year-old singer said he met Smith, 67, for the first time outside a studio in London which he had attended to record vocals for the record, set to be released on July 10.
Jagger said he had noticed a "bloke standing there with his back to me with this long gown on", adding that when he turned around he was "covered in lipstick".
The Stones singer said he told Smith, "You're Robert Smith of The Cure", adding: "While you're here you better go and do something. That's how collaborations work sometimes."
The new album, which follows 2023's Hackney Diamonds, will also feature appearances from Paul McCartney, Steve Winwood, late Stones drummer Charlie Watts and Red Hot Chilli Peppers drummer Chad Smith.
Jagger and guitarists Ronnie Wood, 78, and Keith Richards, 82, spoke to American TV host Conan O'Brien at the album's launch event on Tuesday.
An Instagram post from the band said the record would be "rooted in blues, country, rock and classic Stones songwriting".
Foreign Tongues, recorded over the course of a month in London, will be released on July 10.
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