The bombers detonated explosives-rigged vehicles, one of them a lorry, at three successive checkpoints at the Turaibil crossing, said Sabah Karhout, the head of the Anbar province council.
The deputy head of the crossing, Khaled Alwan, and army Colonel Jabr Khailan, both confirmed the attack took place, with the officer blaming it on the ISIS group.
Iraqi security forces control pockets of territory in Anbar, a vast desert province that stretches from the borders with Syria, Jordan and Saudi Arabia to the western approaches to Baghdad.
Some areas have been out of government control. ISIS seized parts of Ramadi and all of Fallujah, to its east, in early 2014, AFP reports.