The incident happened minutes after the 31-year-old fell out of a sugar apple tree, police sergeant Kosmas Jalang said.
Mr Anwar was left bleeding badly from bites to his hands, body, legs and neck after the two waiting dragons attacked.
He died at a clinic on the neighbouring island of Flores soon afterwards, his neighbour Theresia Tawa said.
Attacks on humans by Komodo dragons are rare but appear to have increased in recent years.
An eight-year-old boy died after being mauled in 2007, while a park ranger survived a similar attack last month.
The latest mauling came as a zoo in Indonesia celebrated the birth of 32 Komodo dragons.
All of the babies hatched in the last two weeks, while another 14 eggs are still incubating at Surabaya Zoo.
Spokesman Agus Pangkat said it was the Komodo’s most successful breeding year at the zoo.
Komodo dragons can grow up to 10 feet long and weigh as much as 150 pounds.