Prime Minister Tarique Rahman, whose term will last the next five years, is the son of former prime minister Khaleda Zia and former president Ziaur Rahman.
He is also Bangladesh's first male prime minister in 35 years.
Since 1991, when Bangladesh returned to democracy, either Rahman's mother or her arch-rival Sheikh Hasina had served as prime ministers.
The country's figurehead president, Mohammed Shahabuddin, administered the oath of office for Rahman on Tuesday.
Dozens of cabinet members and members of the new government were also being sworn in.
The Bangladesh Nationalist Party and its partners won 212 seats in the 350-member parliament, while an 11-party alliance led by the Jamaat-e-Islami party, the country's largest Islamist party, won 77 seats to be the opposition.
A new party - the National Citizen Party, or NCP - formed by the student leaders who led the 2024 uprising was part of the 11-party alliance led by Jamaat-e-Islami.
The NCP secured six seats.
In Bangladesh, voters elect 300 members of parliament directly while the remaining 50 posts are reserved for women and distributed proportionately among the winning parties.
Rahman, 60, who returned to the country in December - after 17 years in self-exile in London and shortly before his mother's death - has promised to work for democracy in Bangladesh, a country of 170 million people.
An interim government led by Nobel peace laureate Muhammad Yunus that took over after Hasina was toppled, oversaw the election.
The vote was largely peaceful and deemed as acceptable by international observers.
Earlier on Tuesday morning, head of the election commission ANM Nasir Uddin administered the oath of office separately to all the newly elected MPs.
Rahman's main rival Bangladesh Awami League party headed by Hasina - who was ousted in the 2024 mass uprising - was banned from the race.
The Yunus-led administration had also banned all activities of Hasina's party, which had ruled the country for 15 years.
From her exile in India, where she has lived since August 5, 2024, Hasina slammed the vote as unfair to her party, which still remains a major political force.
At home, Hasina was sentenced to death on charges of crimes against humanity because of hundreds of deaths involving the uprising.
She denied the allegation and termed the court as a "kangaroo court".
