The attack at 6:40 a.m. on Christmas Day sent hundreds of diplomats and aid workers based at the embassy scrambling into fortified bunkers, the New York Times reported.
Afghan officials said two other rockets struck other parts of the city and three police officers were injured when they attempted to diffuse one that landed without exploding.
Whether the Taliban actually intended to attack the embassy was in question, the newspaper said, noting rocket attacks on Kabul have been infrequent in recent years and rarely appear to be well-targeted.
Taliban spokesman Zabiullah Mujahid posted a Twitter message claiming the embassy sustained heavy causalities. The message contained no reference to Christmas.
The embassy said it was assessing damage to the well-fortified compound.