Al Essawi's killing is likely to further inflame tensions, with Iraq already grappling with a political crisis pitting the prime minister, Nouri Al Maliki, against Essawi's secular Sunni-backed Iraqiya bloc.
Al Essawi, 37, had been inspecting a road being paved south of Fallujah when the attacker, who was dressed as a construction worker, came up to him.
"The moment he stepped out of the car to check out this road between Fallujah and Amriyah, at this moment, there was a man," said Sohaib Haqi, the politician's office chief. "He came to him, hugged him, said 'Allahu Akbar' (God is greatest), and blew himself up."