A separate bombing 30 minutes later at the entrance to the nearby provincial council offices.
It was in this bombing that Abid was injured as he came out of his office to inspect the damage, a source at a Ramadi hospital was reported as saying.
"A suicide bomber wearing an army uniform ran towards the governor," Captain Ahmed Mohammed al-Dulaimi, a local police officer, said.
"Some security people held him back, and he detonated himself."
State television briefly reported that the governor had been killed in the blast, but those reports were quickly denied by Hikmet Khalaf, his deputy.
The AFP news agency quoted a doctor at Ramadi General Hospital as saying: "The governor is wounded. American forces came and took him for more treatment."
The US military did not immediately confirm that its troops took Abid to a US-run hospital.
Ahmed Rushdi, an independent journalist in Baghdad, told Al Jazeera: "There is now a curfew inside Anbar - [the roads] are only for police cars and ambulances. All the members of the council and the governorate have mild injures."
Anbar province was the heart of Iraq's Sunni uprising following the US-led invasion of Iraqi in 2003 but it became relatively secure after local tribal fighters accepted US-backing in 2006.
But a spate of recent attacks has raised fears that violence will increase ahead of Iraq's general elections in March 2010.