Alexander, who played the much-loved character Hilda Ogden between 1964 and 1987, died in hospital three days after her 90th birthday.
She had been admitted to Southport hospital for tests but was discharged after a couple of days and allowed to return to her nursing home, her niece Sonia Hearld, 64, said.
But on Tuesday Alexander was readmitted to hospital, where she later died peacefully on Friday.
Her family said she had recently undergone tests and was readmitted on her birthday after “feeling a little bit poorly”.
Julie Goodyear, who played former Rovers Return landlady Bet Lynch, said Alexander was “an absolute joy to play scenes with”.
She added that she “was the exact opposite of the character she played”.
“This proves what an amazing actress she was.
“Jean was a very private person and meticulous in her work. She really was a perfectionist and always word-perfect.
“Rest in Peace, Jean. Hilda will be remembered forever as one of the best characters in Coronation Street.”
Bill Roache, who has played Ken Barlow in the soap since its first episode in 1960, described working with Alexander as “wonderful”, adding that she “created one of the most iconic Coronation Street characters”.
“She could make you laugh, particularly with Hilda’s comedy singing, but she could also deliver incredibly moving performances and make you cry.”
Michael Le Vell, who plays Kevin Webster, lodged with Hilda with his on-screen wife Sally (Sally Dynevor) after arriving in Weatherfield in 1986. He said starring alongside Alexander was his “proudest and most exciting moment on the show”.
He said Alexander was always first to “make a new arrival feel welcome”.
“She was an all-round lovely lady and such a talent. I will never forget the scenes I had with her and what she taught me.
“She was also great fun. We used to watch snooker in the green room with her as she loved snooker.”
Dynevor said: “Michael and I loved her and any scenes we did with her we always wanted to do our very best.
“Hilda will always be my favourite Corrie character.”
Liverpool-born Alexander won the nation’s heart with her turn as the pinny-wearing, sharp-tongued cleaning lady who was rarely seen without a set of tightly-wound hair curlers.
Fans tuned in in vast numbers to watch her daily battle to get her inept husband, Stan, out of the pub and into work.
Sir Laurence Olivier was among her followers and a host of famous faces including Sir Michael Parkinson, writer Willis Hall, former Poet Laureate Sir John Betjeman and Russell Harty also formed themselves into the British League for Hilda Ogden.
When Alexander decided to leave the soap in 1987, nearly 30 million people watched her farewell episode, according to reports.
In the years after leaving the cobbles she became a mainstay of BBC sitcom Last Of The Summer Wine as eccentric bric-a-brac shop owner Auntie Wainwright.
In 2005 she was voted the greatest soap opera star of all time and there were calls for Hilda to be memorialised.
Alexander said at the time: “I just couldn’t see myself stood on a pedestal cast in bronze.
“Knowing my luck, the Southport seagulls would find me and do their worst, as they have done before - usually when I have just washed my hair.”
Alexander starred in Last Of The Summer Wine until 2010, and she continued to enjoy celebrity status after her retirement in 2012, her niece, Sonia Hearld, told the Press Association.
“People used to stop her just to say hello and how much they missed her,” Mrs Hearld, 64, said.
“She was always very good to her fans - she said ‘They put me where I am’.”
Alexander brought a lot of her own personality to the role for which she will best be remembered, said her niece.
“Certainly with Hilda, the fact she was such a downtrodden person but just got on with it. She just touched a chord with people.
“She did bring a lot of herself to the role but she was a different person, a private person. They were amalgams of people she had met, people she had observed and people she knew. She was very observant about people.”
Despite being best known for her portrayal of downtrodden and impecunious characters, Alexander was remembered as being glamorous off screen.
“Privately, she was great fun and had a great sense of humour,” Mrs Hearld said.
“She was very glamorous - I remember when I was about four or five seeing her get ready for an evening, wearing really pretty clothes and high heels. She occasionally let me traipse around the house in them.
“She was a good role model and she had a drive that carried her through.”
The actress had no children of her own and Mrs Hearld is one of her few surviving relatives.
Alexander reportedly suffered a stroke in 2014 and had been living in a nursing home. Her death appears to have been unexpected.
Mrs Hearld said: “I saw her last week when we went to see her in hospital. She was feeling a little bit poorly but she was talking to me and my husband quite happily.
“There was nothing telling us anything was wrong - she was just generally weak and under the weather.
“We have not had the details.”
John Whiston, creative director at ITV Studios, said: “Everyone who loves what is special about Coronation Street will know how sad it is to hear that Jean Alexander has died.
“Hilda Ogden was the earthy heart of the show for very many years, and the brilliance of Jean as an actor was that she could shift from high comedy to deeply emotional and moving drama in a heartbeat.
“That takes real skill and real acting but also real humanity. That is what the world has lost with her passing.”
A Coronation Street spokeswoman confirmed that Monday night’s episode will be dedicated to Alexander.
They added that show bosses were considering having characters reference the death of Hilda, who would now be 92.