At least two US marines died when militants attacked the perimeter of the huge base in Helmand.
Taliban spokesman Qari Yousef Ahmadi told the BBC that the target was US and UK soldiers and involved 10 insurgents.
Violent protests against Western embassies have swept the Muslim world amid widespread anger over the film.
Protests against the video - Innocence of Muslims - began on Tuesday in Egypt. On Friday, at least seven people died in escalating unrest in Khartoum, Tunis and Cairo.
Nato officials say insurgents used small arms, rockets and mortars in the attack on Camp Bastion.
The sprawling camp is home to troops from several countries and Friday's attack targeted the US compound, Camp Leatherneck.
Maj Martin Crighton from Nato's International Security Assistance Force (Isaf) told the BBC troops were conducting an assessment to determine the extent of the damage.
The video - produced in the US - depicts the Prophet Muhammad as a womaniser and leader of a group of bloodthirsty men. It has been circulating on YouTube.
Camp Bastion has a high level of security and is one of the world's busiest airports because of the huge number of helicopter and aeroplane flights landing and taking off.
The UK's Prince Harry is currently based there, on his second Afghan tour of duty.
Nato told Reuters that the prince was on the base at the time of the attack but was "never in any danger".