At least 19 others are reported to have been injured in protests across the country, as anger over the cartoons shows no signs of abating.
In the central Afghan city of Mihtarlam, police fired on the demonstrators after a man in the crowd shot at them and
others threw stones and knives, said Dad Mohammed Rasa, a spokesman for the Interior Ministry.
The protesters burnt tyres and threw stones at the offices of the police and provincial governor.
Meanwhile in the capital, Kabul, about 200 protesters tried to break down the gate of the Danish government's diplomatic mission, said police who were guarding the building.
The protesters then threw stones at the mission and attacked officers guarding it, as well guards at a nearby house used by Belgian diplomats.
In Indonesia, the world's most populous Muslim nation, protesters hurled rocks and broke windows of a Danish consulate in the country's second largest city, Surabaya, witnesses said.
Demonstrations were also held in two other Indonesian cities, including Jakarta, as well as in Thailand, Kashmir and in several other parts of Afghanistan.
Protesters attacked the Danish
government's diplomatic office
In the southern Afghan city of Kandahar, more than 100 people marched through a bazaar in the former Taliban stronghold chanting "Death to Denmark" and carrying flags inscribed with "God is great".
A similar protest gathered in the capital, Kabul.
Indonesian Muslims staged noisy but peaceful protests in four cities demanding Denmark apologise over the controversial cartoons.
The Indonesian protests happened as the editor of a Malaysian newspaper resigned for publishing one of the cartoons and Muslim organisations demanded an Australian newspaper apologise for reprinting one.
About 200 protesters from a leading Islamist party rallied near a building housing the Danish embassy in Jakarta on Monday.
The embassy is on the 25th floor and the flag-waving demonstrators from the Prosperous Justice Party (PKS) were barred from entering the lobby, where unruly protesters from a hardline group rampaged on Friday.
Muslims have protested all over
the world against the cartoons
The protesters shouted slogans condemning the caricatures, which were first published by Danish daily Jyllands-Posten last year.
"Denmark must apologise for disgracing the prophet," yelled the protesters.
One banner read: "Insulting the Prophet = Insulting Islam".
There were also small protests by separate groups in three other cities. There was no violence.
Meanwhile an editor of a small Malaysian newspaper resigned on Monday for reprinting the cartoons.
The Sunday Tribune newspaper in the remote Borneo state Sarawak released a statement apologising and expressing "profound regret over the unauthorised publication" of one of the drawings by an editor on duty.
The statement also was published by the newspaper on its front page on Sunday, but did not get much publicity in Malaysia because of the remoteness of the state.
The Sunday Tribune, which carried the drawing on its page 12 on Saturday, is the only newspaper in the mainly Muslim country to have reprinted any of the caricatures.
It added that the editor, who was not identified, has admitted and regretted his oversight and officially written an apology. The editor "has voluntarily resigned forthwith", the statement said.
A protest had been organised in Kuala Lumpur on Friday but less than 100 people turned up.
An earlier round of protests was
staged in Indonesia on Friday
In the Australian city of Canberra, outraged Muslims on Monday demanded a newspaper apologise after it published one of the cartoons.
The News Corp-owned Courier-Mail, the biggest newspaper in Queensland's state capital Brisbane, apparently became the first newspaper in Australia to publish one of the Danish caricatures on Saturday despite warnings from Muslim groups.
But the low-key cartoon that illustrated an article on page 17 about an attack on the Danish embassy in Jakarta by angry Muslims did not catch national attention until Monday.
Abdul Jalal, the Islamic Council of Queensland chief, said the paper should apologise to the state's Muslims.
Jalal told Australian Broadcasting Corporation radio: "I was hoping, praying that our media people would have more ... sense, in not trying to agitate the situation in the local scene here in Australia."
Courier-Mail editor David Fagan declined to comment.
"The role of the government in free speech matters is to do nothing"
Australian president, IFJ
Chris Warren, the Australian president of the International Federation of Journalists (IFJ), said there was no place for government intervention.
Warren said: "The role of the government in free speech matters is to do nothing."
Outrage has erupted in the Middle East and other countries in the Muslim world after more European newspapers published the cartoons, which were originally published last September.
Muslims consider any images of Prophet Muhammad to be blasphemous.