DUBAI (Reuters) - Hundreds of Yemeni demonstrators stormed the U.S. embassy in Sanaa on Thursday in protest at a film they consider blasphemous to Islam, and security guards tried to hold them off by firing into the air.

The attack followed Tuesday night's storming of the United States Consulate in Benghazi, where the ambassador and three other staff were killed. President Barack Obama said the perpetrators would be tracked down and ordered two destroyers to the Libyan coast, but there were fears protests would spread to other countries in the Muslim world.

Young demonstrators shouting "we redeem, Messenger of God" smashed windows of the security offices outside the embassy with stones and burned cars before breaking through the main gate of the heavily fortified compound in eastern Sanaa. Others held aloft banners declaring 'Allah is Greatest'.

Tyres blazed outside the compound and protesters scaled the walls.

"We can see a fire inside the compound and security forces are firing in the air. The demonstrators are fleeing and then charging back," one witness told Reuters.

Witnesses said there were some injuries on both sides but gave no exact figures.

Yemen, a key U.S. ally, is home of Al Qaeda in the Arabian Peninsula (AQAP), which is viewed by Washington as the most dangerous branch of the militant network established by Osama bin Laden.

Yemen is fighting an al Qaeda-backed insurrection largely in the south of the country.