The former Bond actor, 88, posted the news alongside photographs of her on Twitter.
Knudsen was the daughter of Sir Roger's fourth wife, Kristina "Kiki" Tholstrup.
Thought to be aged 47 at the time of her death, Knudsen was previously married to investment banker Nikolaj Albinus.
The couple's high-profile wedding in the south of France was featured in Hello! magazine, but their marriage lasted only 13 months.
She had lived in London since 1997 and had worked in interior design and property development.
In 2014 she launched Ochun, a private social club for single people.
She told the Evening Standard: "After I divorced I realised how difficult it was to meet like-minded people who were single. I was in my early 30s.
Following her divorce, she had a long-term relationship with Skype co-founder Janus Friis.
Her father, businessman Hans Christian Knudsen, committed suicide, and many years later her stepfather Ole Tholstrup died from alcoholism.
She was very close to Sir Roger and is quoted as crediting him with "turning her life around".
Wildlife activist and author Lauren St John was among the people who sent Sir Roger a message of condolence on Twitter.