Danesh was found dead in his apartment in Rochester, Minnesota, on 11 August, his family announced.
The cause of death is unknown at this time, however the local police department confirmed there were no suspicions of foul play.
In a statement given to the PA news agency, former Pop Idol judge Cowell, 62, said: “I first saw Darius on television over 20 years ago and I got to know him really well.
“He was charismatic, funny and just a great person to be with.
“His passing, for someone so young, is an absolute tragedy and my heart goes out to his family and friends.”
In 2002, Danesh came third on Pop Idol behind Will Young and Gareth Gates. His memorable moments on the show include his renditions of Tom Jones’s hit “It’s Not Unusual” and Atomic Kitten’s “Whole Again”. The final was watched by more than 13 million viewers.
Having turned down a record deal from Cowell, Danesh nevertheless reached number one in the UK later that year with his track “Colourblind”. His album Dive In reached the top 10.
Danesh later became a West End star, appearing in musicals such as Chicago and the Oliver Award-winning Guys and Dolls.
Danesh said he based his portrayal of Chicago’s scheming lawyer Billy Flynn on his old mentor and Pop Idol judge, Cowell.
Danesh starred opposite Ugly Betty star America Ferrera in the musical set in prohibition-era Chicago, and said: “He’s the consummate businessman and a wonderful guy and there’s a lot of Simon Cowell in Billy Flynn in terms of the way he approaches things.”