He was taken to a hospital for hospice care late last week and died on Saturday with his wife and brother at his side, it said.
Craft was the director of the New York Asian Film Festival and spoke fluent Mandarin.
He had endured more than three years of cancer treatments and thought he had beaten the disease before being diagnosed with a separate terminal tumour just six weeks ago.
His wife said he had enjoyed Star Trek Into Darkness "immensely" and they were honoured Abrams took the time to arrange a viewing, in a recent Reddit post.
"It was truly amazing that a film-maker so secretive as JJ Abrams was kind enough to show this to us...
"It was a wonderful thing to see with Daniel and a wonderful thing to see Daniel enjoy - Making someone as ill as he is smile for any length of time really makes a difference. (At this point making me smile is good too)."
The woman said she was sworn to secrecy but it was poignant Abrams was concerned they wouldn't enjoy the film because it was still being worked on.
"Be assured we enjoyed it 110 per cent.
"We are so grateful thank you, thank you, thank you."
- Herald Online