The RTE broadcaster died last Thursday at the age of 69.
Her funeral is taking place at St Brigid’s Church in Kill before a private burial.
Those attending the service included Taoiseach Leo Varadkar, Fianna Fail leader Micheal Martin, as well as RTE colleagues and broadcasters Joe Duffy, Clare Byrne, Ryan Tubridy, Brendan O’Connor, Miriam O’Callaghan and RTE’s Director General Dee Forbes.
Government ministers including Minister for Children, Katherine Zappone, and Minister for Communications Richard Bruton also attended.
The chief celebrant of the mass Monsignor Ciaran O’Carroll said the news of Finucane’s death was a “dreadful shock”.
“She was such an icon of Irish broadcasting and will always be remembered for her outstanding journalistic abilities, her keen intellect, warmth, wit, thoughtfulness, compassion and kindness,” he said during the service.
“The influence of Marian was significant.
“The contribution Marian made to advancing and shaping our nation and its peoples through her professional work as a courageous broadcaster, and her determination to facilitate conversation without fear or favor, was repeatedly and wonderfully captured in countless public tributes paid over recent days.
“We have also come here this morning to offer our sympathy and express our solidarity with all the members of her family – especially her husband John, her son Jack, her daughter-in-law Jenny, her sisters, brothers, in-laws, family members, relatives, colleagues and friends.”
Thousands always enjoyed her programs and she will be missed on the radio. Publicly the nation has lost a skilled broadcaster.
He added that the nation has lost a “skilled broadcaster”.
“She had great empathy and sense of fairness and treated everyone with courtesy and respect,” Monsignor O’Carroll added.
“A generous and determined woman, and a thoughtful and kind friend whose keen sense of humor enriched the lives of many, her charity work at home and abroad was simply outstanding.
“Thousands always enjoyed her programs and she will be missed on the radio. Publicly the nation has lost a skilled broadcaster.
“Privately John has lost a soul partner, Jack a loving mother and Jenny a cherished mother-in-law.
“The great, exuberant joy shared by our family and friends but a few short months ago at Jack and Jenny’s wedding contrasts now with the sorrow and sadness we experience this day of Marian’s funeral.”
He said the broadcaster was blessed with an “indomitable spirit” which she had to call on often in her full, dynamic and fruitful life.
“Blessed with energy and dynamism, determined and persistent, she had boundless curiosity about life and the world,” he continued.
“Throughout her eventful life she was privileged to experience many changes and received the grace to confront various, sometimes demanding, challenges.
“She had to carry the cross of bereavement with the death of her beloved daughter Sinead – a bereavement she bore with tremendous dignity.
“In the face of Sinead’s death, Marian mobilized all of her formidable strengths and skills by dedicating a whole chapter of her life to supporting the hospice movement at home and establish a hospice charity in South Africa.
“Through her various popular programs Marian became a presence in thousands of homes each week, as listenership figures consistently revealed.”