Anti-riot police kept protesters from entering the embassy on Tuesday evening, but this did not prevent demonstrators from throwing Molotov cocktails over the wall and into the garden.
There were no clashes or arrests, and the extent of any damage was not immediately known. The demonstrators were demanding the closure of the mission.
Excerpts of the video, which was originally published on the internet, were broadcast on Denmark's TV2 channel.
They showed members of the extreme-right Danish People's Party (DPP) portraying the Prophet Muhammad as a beer-drinking camel and a drunken "terrorist" attacking Copenhagen.
The video was filmed by Martin Rosengaard Knudsen, a member of an anti-racist group called Defending Denmark, who infiltrated the DPP to expose its racism.
Danish embassies and commercial interests in Muslim countries were hit by a wave of violent protests in January and February this year after satirical cartoons of the Prophet Muhammad were published in a Danish daily, Jyllands Posten, in September 2005.