The performer, who was best known for her work on the band’s critically acclaimed 1991 album Screamadelica, died “suddenly at home” in Manchester.
A statement from her family said Denise had been “ill in the week prior to her death, but told friends she was ‘much better’ on Friday”.
Denise’s cause of death has not yet been disclosed.
The singer performed with Primal Scream between 1990 to 1995, featuring on one of their biggest songs, Don’t Fight It, Feel It.
Denise also featured on records from some of her home town’s biggest bands, including New Order, as well as performing on tracks by The Pet Shop Boys.
She had sung with Manchester band A Certain Radio for over 25 years.
New Order’s official Twitter account said the band was “devastated” by Denise’s death.
A statement said: “She was a beautiful person with a huge talent. Her voice graced many of our albums and live shows and that of our friends. We will miss her dearly.”
Peter Hook, former bassist of New Order, also said on Twitter: “Shocked and saddened to hear about Denise – my heart goes out to the Johnson family for their terrible loss. Rest in peace Denise, singing with the angels. X.”
Tributes to Denise poured in from across the music industry, including from Happy Monday’s Rowetta, former Smiths guitarist Johnny Marr, Stone Roses lead singer Ian Brown and Boy George.
Denise’s solo acoustic album, Where Does It All Go, is set to be released in September.