Majid al-Assafi, the head of the province's police force, said: "Our troops are on alert to avert any failure of the transfer."
Before the handover, US forces increased their patrols in Ramadi, the provincial capital.
There are 28,000 US soldiers in Anbar at present, down from February when the US put the figure at 37,000.
Three years ago there were 5,000 Iraqi soldiers, a deployment that has now grown to 37,000.
Reducing troop numbers in Iraq will allow the US to redploy to Afghanistan, a move encouraged by Democratic presidential hopeful Barack Obama.
Some US troops will remain and will carry out training as well as participate in military operations when required.
Iraq and the US have so far failed to agree a timetable for withdrawal, with some critics remaining unconvinced that a handover will be successful.
One resident said: "Our forces are not ready to take responsibility for the situation. There is a lot of corruption in our security forces."
US and civilian casualties started to decline in the province after the creation of so-called "Awakening Councils", local tribal groups who are funded by the US.
The handover was originally due on June 28.