Glenys Kinnock died peacefully in her sleep on Sunday with her husband and former Labour Party leader Lord Kinnock by her side, her family said.
She was raised in Holyhead and went on to serve as a minister in the New Labour government and also represented Wales in the European Parliament as an MEP.
Her family, which includes Labour MP Stephen Kinnock and daughter Rachel, said they were "devastated" by her death.
Former Ynys Mon MP Albert Owen said: "Sad to hear the passing of Glenys Kinnock - a lovely, clever and caring person. Proud of her Holyhead roots, and attending the 1st ‘Comp’ school.
"A committed internationalist and best campaigner I’ve known. Condolences to Neil, Stephen, Rachel and family."
Labour leader Sir Keir Starmer called her a "true fighter" for the party as he paid tribute to her life and career.
The 79-year-old, Baroness Kinnock of Holyhead, had been diagnosed with Alzheimer's six years ago.
In a statement to PA news agency, her family said: "It is with the deepest sorrow that we announce the death of Glenys Kinnock. Glenys died peacefully in her sleep in the early hours of Sunday morning, at home in London.
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"She was the beloved wife and life partner of Neil, the cherished mother of Steve and Rachel and an adored grandmother. Neil was with her in her final moments. They had been married for 56 years.
"A proud democratic socialist, she campaigned, in Britain and internationally, for justice and against poverty all her life."
Baroness Kinnock served as an MEP for some 15 years before leaving Brussels in 2009 to take up a life peerage when then prime minister Gordon Brown appointed her minister for Europe.
In their statement, the family continued: "Passionate to the end about education, she was a valued and respected school teacher before she began her own political career, as a member of the European Parliament, then being made a peer in the House of Lords from where she served as minister for three of the great passions of her life, Europe, Africa and the UN.
"She was a great friend to many people and causes and was truly loved.
"Glenys endured Alzheimer's after being diagnosed in 2017 and, as long as she could, sustained her merriment and endless capacity for love, never complaining and with the innate courage with which she had confronted every challenge throughout her life.
"The family is of course devastated and would ask that their privacy be respected. Funeral details will be communicated in due course."
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Sir Keir said Baroness Kinnock was a "passionate lifelong campaigner for social justice at home and abroad" who had an "impressive political career" in her own right.
"Neil and Glenys had the most wonderful partnership, there for each other through thick and thin, with a love and commitment that was instantly obvious when you saw them together," he said.
"As the family have detailed, in recent years that meant looking after Glenys as Alzheimer's did its worst.
"But what we will all remember is Glenys as a true fighter for the Labour Party and the values of the labour movement, a pioneering woman, to whom we owe an enormous debt.
"My sincere condolences to Neil, Stephen, Rachel and all the family at this sad time."