The 27-second video was apparently taken with a mobile phone camera.
The footage begins by showing a body on a stretcher covered in a white shroud and the camera moving upwards.
As the camera reaches the head, the shroud is pulled aside to reveal the bearded face of Saddam twisted sharply to his right with a wound just below his jaw on the left side.
A few red blotches are also seen on Saddam's left cheek. Voices can be heard, apparently of those shooting the film.
As the shroud is pulled back, one voice is heard saying: "Hurry up, hurry up. I will count one to four."
Then another voice is heard saying: "Abu Ali, you take care of this."
An earlier video, posted on the internet a day after the hanging and also shot on a mobile phone camera, showed Saddam at the gallows being taunted as guards in black balaclavas slipped the noose around his neck, moments before the trap-door opened.
The British prime minister's first remarks about the hanging on December 30 came at a joint press conference with his Japanese counterpart, Shinzo Abe, on Tuesday in London.
Blair said he hoped that disputes over the taunting and the release of illicit video footage of the execution would not lead people to forget the gravity of Saddam's crimes.
"The crimes that Saddam committed does not excuse the manner of his execution, but the manner of his execution does not excuse the crimes," he said.
"As everybody saw, the manner of the execution is unacceptable and it's wrong, but we should ... not allow that ... then to lurch into a position of forgetting the victims of Saddam, the people that he killed deliberately."
Blair had faced public criticism after choosing not to immediately comment on the execution.
In other news, Iraqi officials have handed a man over to magistrates after a government investigation into who taunted Saddam with sectarian abuse moments before he was hanged, an Iraqi official said on Tuesday.
Ali al-Dabbagh, an Iraq government spokesman, said an official government inquiry concluded that the taunting could be blamed on "individual behaviour".
Also on Tuesday, Nuri al-Maliki, the Iraqi prime minister, confirmed that Zalmay Khalilzad, the outgoing US ambassador to Baghdad, had requested a 10-day or two-week postponement of Saddam's execution.
"There was a wish expressed by the US ambassador that the execution be postponed by 10 days or two weeks," he said in a TV interview.
The request was rejected to avoid provoking demonstrations by "those who were angry with Saddam", al-Maliki said.