The shooting came as protesters, chanting "O, messenger of Allah... O, Mohammed," launched a second charge on the complex which they had stormed earlier but were ejected by the security forces.
President Abdrabuh Mansur Hadi apologised to his US counterpart Barack Obama and the American people for the acts of a "mob" and ordered a probe.
"Those who are behind (the attack) are a mob that are not aware of the far-reaching plots of Zionist forces, especially those who made a film insulting the prophet," said Hadi.
Some protesters said they saw three vehicles being torched by some of the demonstrators after they gained access to the compound through an unguarded security gate.
After being evicted from the complex on their first assault, protesters retreated about 100 metres from the gate, gathering near a checkpoint where they chanted anti-Jewish slogans.
They then launched a second bid to access the compound, prompting police to fire on the crowd, witnesses said.
The violent protests in Sanaa come two days after four Americans, including the ambassador, were killed when a Libyan mob attacked the US consulate in Benghazi, and protesters in Cairo tore down the Stars and Stripes and replaced it with a black Islamic flag.