The military operation to liberate the city 40 miles west of Baghdad began on Monday. At least 20,000 troops have been deployed to combat jihadists who were said to trap 90,000 Fallujah residents.
"A total of 2,030 people, or 507 families, have been helped out through safe corridors," the command said as quoted by the Al Sumaria broadcaster, Sputnik reported.
Some 110 Daesh militants have been killed, their vehicles and bomb-making facilities destroyed, the command added. Previously, at least 80 Daesh militants have been killed west of Fallujah.
The United Nations estimates only a small number of Fallujah’s population was able to escape the city, which became an Islamist stronghold in 2014. Some 800 people from mostly outlying areas have fled, with 50,000 still inside, the UN estimates.