The compound houses the offices of Safer oil company, the former Hunt Oil firm which is now owned by Yemen.
Yemeni security officials confirmed that rockets had been fired but gave few other details.
One security official was quoted by state media as saying there were no casualties.
Residents told the AFP news agency that three rockets struck near the villas where American oil experts from Safer live.
In a statement posted on an Islamist website, the Jund al-Yemen Brigades said it "launched five mortar shells at the American embassy in Sanaa".
The US-based SITE Intelligence Group, which monitors such websites, said the statement did not give a date for the attack but acknowledged that one shell missed the target and hit the neighbouring school.