With about 128 million people eligible to vote, turnout reached a high 33 per cent by noon on Thursday at three-quarters of polling centres across the country, Akhtar Ahmed, senior secretary of the Election Commission, told reporters.
Analysts say a decisive result is crucial for steady governance in the nation of 175 million, as the deadly anti-Hasina protests triggered months of unrest and disrupted key industries, including the huge garments sector, the world's second-largest exporter.
It is the world's first election after an uprising led by under-30s, or Gen Z, to be followed by Nepal in March.
The contest pits two coalitions led by former allies, the Bangladesh Nationalist Party (BNP) and the Islamist Jamaat-e-Islami, with opinion polls giving an edge to the BNP.
In the capital Dhaka, people queued up outside voting booths before polls opened, including eager participants such as Mohammed Jobair Hossain, 39, who said he last voted in 2008.
"I am feeling excited because we are voting in a free manner after 17 years," Hossain said as he waited in line.
"Our votes will matter and have meaning."
Hossain's sentiment was echoed by many voters, who told Reuters the atmosphere felt more free and festive than earlier elections.
Kamal Chowdhury, 31, who works as a driver for a company in Dhaka and travelled to his hometown in the eastern district of Brahmanbaria to cast his vote, said: "It feels festive here.
"People are so enthusiastic to cast their vote - it's almost like Eid," he said, referring to Eid al-Fitr, the Muslim religious festival.
Outside a polling booth in Dhaka where BNP chief Tarique Rahman and head of the interim government Muhammed Yunus voted, policemen were on horses with saddle blankets proclaiming: "Police are here, vote without fear."
Hasina's Awami League is banned, and she remains in self-imposed exile in long-term ally India, opening the window for China to expand its influence in Bangladesh as Dhaka's ties with New Delhi deteriorate.
Elections were held during Hasina's tenure but they were marred by opposition boycotts and intimidation, critics have said.
This time more than 2000 candidates, including many independents, are vying for 300 seats in the Jatiya Sangsad, or House of the Nation.
The two prime ministerial candidates are the BNP's Tarique Rahman and Jamaat chief Shafiqur Rahman. They are not related.
"I am confident of winning the election. There is enthusiasm among the people about the vote," Tarique Rahman told reporters.
Voting in one constituency has been postponed due to the death of a candidate.
At least 50 parties are contesting in total, a national record.
"From today, we have the opportunity to build a new Bangladesh with every step we take. This is a festival, a day of joy, a day of liberation, the end of our nightmare. I congratulate you all," Yunus, who took over as interim head after Hasina, said after voting.
About 958,000 personnel from the police, army and paramiltary forces were deployed throughout the country on election day, the Election Commission said.
Police and army personnel were stationed outside most polling booths.
Alongside the election, there will be a referendum on a set of constitutional reforms, including increasing women's representation, strengthening judicial independence and imposing a two-term limit on the prime minister.
Counting will begin soon after polling closes, with early trends expected about midnight and results likely to be clear by Friday morning.
