The BBC reports that Sullivan, who was 64, had suffered from viral pneumonia, and had spent six weeks in intensive care at a hospital in Surrey.
Speaking of the news, BBC director general Mark Thompson said: "John had a unique gift for turning everyday life and characters we all know into unforgettable comedy.
"His work will live on for years to come. We will miss him and we send our condolences to his family."
Gareth Gwenlan, producer of Only Fools and Horses and a close friend, said: "The sudden death of John Sullivan has deprived the world of television comedy of its greatest exponent.
"John was a writer of immense talent and he leaves behind him an extraordinary body of work which has entertained tens of millions of viewers and will continue to do so for many decades to come.
"I have had the privilege of working with John for over 30 years as a colleague and close friend. He was a writer at the peak of his creative powers with so much more to give."
Mark Freeland, BBC head of comedy, added that he was "the Dickens of our generation. Simply the best, most natural, most heartfelt comedy writer of our time."
Sullivan, who was awarded the OBE in 2005 for services to drama, was married with three children and two grandchildren.
He also wrote Citizen Smith, and his most recent work Rock & Chips is due to air on BBC One on Thursday.
Watch a classic clip from Only Fools and Horses below:
> Remembering... John Sullivan, 1946-2011