The husky-voiced singer, known for a string of hits including Total Eclipse of the Heart, died in Portugal last month aged 75.
More than 1,000 people gathered outside the church ahead of the funeral in south Wales, which was also live-streamed.
There was applause as a hearse carrying her coffin – adorned with white roses and lilies and etched with a small hand holding a microphone and a treble clef – arrived at Swansea Minster, packed with family and friends.
It’s a Heartache and another of her hits, If I Sing You A Love Song, were played as pallbearers carried the casket into the minster.
“She is a Welsh icon,” 75-year-old retiree Bette Richards told AFP. “She was a beautiful person. She never changed,” she added.
The funeral also featured Simon & Garfunkel’s Bridge Over Troubled Water performed by a male voice choir.
Another Swansea resident, John Edwards, 70, paid tribute to a star who was also very much “an ordinary person” and “one of us”.
“I feel proud … Everybody loves her. It is a big loss,” he said.
