At least 15 people have been reported killed from several districts.
The Bangladesh Election Commission told Reuters it had heard vote-rigging allegations from "across the country" and would investigate.
The election is likely to deliver a third straight term for Prime Minister Sheikh Hasina. Her main rival is serving a jail term for corruption.
The build-up to the poll was marred by violence and accusations of a crackdown against the opposition. Some 600,000 security personnel were deployed to prevent clashes.
The authorities ordered high-speed internet be shut down until after the vote to prevent the spread of "rumours" that might spark unrest.
Minutes before polls opened, a BBC correspondent saw filled ballot boxes at a polling centre in the port city of Chittagong. The presiding officer declined to comment.
Only ruling party polling agents were present at that and several other polling centres in the second largest city of the country.
At least 28 candidates from the main opposition alliance withdrew before polling closed, alleging vote rigging and intimidation.
More than 100 million people were eligible to vote but reports suggest turnout has been low.
The governing Awami League (AL) said that, despite sporadic incidents, voting was held in a festive atmosphere.
The BBC's South Asia Regional editor, Anbarasan Ethirajan says if the opposition claims of widespread vote rigging are proved, then it will likely to raise questions over the credibility of the election process and the legitimacy of the new government.
Bangladesh is a Muslim-majority nation of more than 160 million people and faces issues ranging from possibly devastating climate change, Islamist militancy, endemic poverty and corruption.
The country has recently been in the international spotlight as hundreds of thousands of Rohingya Muslims have fled there from neighbouring Myanmar.
More about: Bangladesh