"No one saw this coming," says Keith Richards in a statement. "Mick's always been my soul brother and we love him. We're as thick as thieves, and we're all feeling for the man."
Ronnie Wood added, "This is such terrible news, and right now the important thing is that we are all pulling together to offer Mick our support and help him through this sad time."
The band was set to begin the Australian and New Zealand leg of their 14 ON FIRE tour on Tuesday, but has postponed all shows following the news of Scott's death.
"Needless to say, we are all completely shocked," says drummer Charlie Watts. "Our first thought is to support Mick at this awful time. We intend to come back to Australia and New Zealand as soon as possible."
Jagger posted a heartfelt tribute to Scott on his Facebook page Tuesday.
"I am still struggling to understand how my lover and best friend could end her life in this tragic way," he wrote. "We spent many wonderful years together and had made a great life for ourselves."
Scott's death was officially ruled a suicide by the New York City medical examiner's office on Wednesday.
An autopsy completed on the body of the fashion designer found that she died of hanging, Julie Bolcer, a spokeswoman for the city medical examiner confirms.
Scott was found dead by her assistant in her Manhattan apartment Monday morning. Police told PEOPLE that she was found kneeling with a scarf wrapped around her neck that had been tied to the handle of a French door. She did not leave a note.
For much more on L'Wren Scott, pick up this week's issue of PEOPLE magazine, on newsstands Friday