The club, credited with launching the career of The Beatles, opened its doors on 16 January 1957 as a jazz cellar.
Crowds gathered in Mathew Street to see the sculpture which shows a young Cilla performing one of her early songs.
The TV celebrity and 1960s singing star, who died in 2015, started work as a cloakroom attendant at the club.
It was commissioned by Black’s sons Robert, Ben and Jack Willis, who said they backed the idea after being moved by the response from the city following their mother’s death.
Robert Willis said they wanted to “donate it as a small gesture of gratitude to this great city for their wonderful outpouring of love and affection for our mother”, adding that he was “thrilled” with the statue.
His mother would have been “very flattered, proud and honoured”, he said.
Andy Edwards, one of the sculptors, said “it’s the story of the birth of that period in Liverpool’s musical culture” and it was important the city remembered her.
The Beatles played hundreds of gigs at The Cavern Club between 1961 and 1963.
The unveiling was one of a programme of special events taking place on Monday to mark 60 years of the Cavern, which became synonymous with Merseybeat and The Beatles.
The Cavern was demolished in 1973 to make way for a shopping centre but reopened 10 years later on part of the same site using reclaimed bricks from the original building.
Cavern director John Keats said the club had remained relevant and people were “constantly surprised at who has played [here]”.
John Lennon’s half sister, Julia Baird, said the role it played in the Liverpool’s musical history should not be underestimated and she thought Lennon “would have loved” to be part of the celebrations.
“The Beatles didn’t launch The Cavern – The Cavern launched The Beatles,” she said.
The club’s owner, Dave Jones, said: “This venue has to be protected for ever… it has to remain here.”