Police shot in the air in an attempt to hold back the crowds, but failed to prevent them gaining access to the compound and setting fire to vehicles.
Security forces have now regained control of the compound.
US officials say they are investigating whether the attack in Libya was planned, citing suspicions that a militant jihadist group may have co-ordinated the violence.
Three other US consul staff and several Libyans died in that attack, along with Ambassador J Christopher Stevens, who is believed to have died from smoke inhalation.
Security force reinforcements in Sanaa used tear gas, water cannon and live fire to drive back protesters.
"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 the Reuters news agency.
The crowd has been pushed out of the compound, but protests are reported to be continuing outside. There were reports of injuries on both sides, although Reuters carried a statement from the embassy saying there were none.
It was not immediately clear whether the embassy was occupied at the time. There are reports that embassy staff have been moved to a safer location.
There have also been clashes over the past three days in the Egyptian capital, Cairo.
On Wednesday, demonstrators in Cairo angry at the film - Innocence of Muslims - breached the walls of the US embassy and tore down the flag. The clashes, which began on Tuesday, continued in the early hours of Thursday morning.