Sarvendra Nath Khanal, Katmandu's police chief, said that the explosion was at Ratna Park, located in the centre of the city and next to the grounds where political rallies are held.
No one has taken responsibility for the blast.
Political leaders have said the assembly will dissolve the 240-year-old monarchy and sack King Gyanendra when it holds its first session on Wednesday.
Wearing suits and jackets, Nepali caps and traditional robes, the legislators, a third of them women and many from minority groups, repeated the oath in 40 different languages.
After abolishing the monarchy and declaring Nepal a federal democratic republic, the legislators are due to draft a new constitution - the next step in the peace process after a civil war that killed at least 13,000 people.
"This is an epoch-making day," Prachanda, a Maoist leader who won his seat in Kathmandu by a landslide and is slated to head the next government, said.
"For the past 50 years people have been fighting for this."
King Gyanendra has so far refused to heed calls that he leave his palace before Nepal is declared a republic on Wednesday, prompting senior officials to threaten that he could be removed by force.
Ram Chandra Poudel, the peace and reconstruction minister, said: "If he does not leave the palace then the government might have to use force."
"This will not be good for him."
The swearing in came a day after two small bombs were detonated outside a Kathmandu convention centre and another was set off outside the home of an anti-monarchy activist.
The attacks caused no deaths or injuries but underscored the challenge the new assembly faces.
The Maoists, who waged a decade-long armed struggle before signing a peace deal in 2006, won more than a third of the constituent assembly's 601 seats in elections in April.