A statement purportedly from the leader of al-Qaeda in Iraq hailsáthe defeat of Republicans in the US mid-term polls.
Abu Hamza was identified in June as the late Zarqawi's successor
The audio message, whose authenticity has not been verified, was published on Islamist websites and was said to be the voice of Abu Hamza al-Muhajir.
The Democrats' victory in Tuesday's Congressional elections was a move in the right direction, the speaker said.
Outgoing US Defence Secretary Donald Rumsfeld had stepped down to flee the Iraqi battlefield, he added.
He told US President George W Bush to "stay on the battleground".
"I tell the lame duck (US administration) do not rush to escape as did your defence minister.
"The American people have taken a step in the right path to come out of their predicament... they voted for a level of reason," the voice said. Muhajir, also known as Ayyub al-Masri, has been identified by US forces as the successor to Abu Musab al-Zarqawi, killed in a raid in June 2006.