Security forces fired tear gas in the capital, Dhaka, to disperse hundreds of stone-throwing protesters who are part of an alliance of about a dozen Islamic groups.
Witnesses say the protesters burned vehicles, including a police van. Some demonstrators were arrested.
Authorities had banned all protests in the area starting late Friday.
The alliance that led the demonstration called for a nationwide strike Sunday to protest the Internet video, which was produced in the United States.
In Nigeria, thousands of people opposed to the video took to the streets of Kano, the largest city in the country's mainly Muslim north. Demonstrators marched toward the palace of the Emir of Kano, the region's top Muslim spiritual leader.
The protests come a day after at least 17 people died in clashes between security forces and demonstrators in Pakistan.
Thousands of Muslims also demonstrated Friday in other countries, including Afghanistan, India, Sri Lanka, Bangladesh, Iraq, Lebanon and Indonesia. Some protesters burned American flags and effigies of U.S. President Barack Obama.
The low-budget Internet video was produced by an anti-Muslim filmmaker in California. It first sparked protests last week in Cairo and the Libyan city of Benghazi, where U.S. Ambassador Christopher Stevens and three other embassy personnel were killed. Since then, anti-U.S. protests have spread as far as Indonesia.
Protesters have also voiced anger about images of the Prophet Muhammad, including some of him naked, that ran in a French magazine a few days ago.
France's government closed embassies, consulates, cultural centers and schools Friday in 20 countries as a precautionary measure. French authorities also banned two anti-U.S. protests planned for Saturday.
Some information for this report was provided by AP, AFP and Reuters.