Images posted online showed vehicles crushed under giant steel girders at the departure forecourt of the airport's Terminal 1, one of several projects formally launched by Modi in March ahead of the country's general election.
The terminal is used for domestic flights only and airport authorities said all flights departing from it had been cancelled until early afternoon.
"Due to heavy rain since early this morning, a portion of the canopy... collapsed around 5 am," they said in a statement posted online.
"Emergency personnel are working to provide all necessary assistance and medical aid to those affected."
Other terminals were operating as normal, as were arrivals at Terminal 1, it added.
"Eight people have been injured, one person is dead. Rescue operations have been completed," Atul Garg, director of Delhi Fire Services told AFP.
Infrastructure spending has been a priority under Modi, and the Delhi airport -- named after his assassinated predecessor Indira Gandhi -- is one of the country’s flagship projects.
But safety and construction standards in the country remain a concern. Last year, more than 40 workers were trapped for nearly two weeks before they were rescued after the road tunnel they were working on in Uttarakhand collapsed.
And at least 18 labourers working on a railway bridge being built across a ravine in the northeastern state of Mizoram were killed when it collapsed.
At the scene, civil aviation minister Ram Mohan Naidu Kinjarapu told reporters: "We are taking this incident seriously...I want to clarify that the building inaugurated by PM Narendra Modi is on the other side and the building that collapsed here is an old building and was opened in 2009."
Modi only took power several years later, but opposition politicians slammed the prime minister after the Friday incident.
A Congress party spokesman said he had gone on an "inauguration spree" ahead of the polls, where Rahul Gandhi's organisation defied expectations and deprived Modi's Bharatiya Janata Party of its overall majority in parliament, forcing it into a coalition with allies.
Gandhi's sister Priyanka -- herself a senior Congress figure -- posted on social media platform X: "Will the Chief Inauguration Minister take responsibility for this poor construction work and this corrupt model?"
A section of roof at Jabalpur airport in Madhya Pradesh -- another of the projects inaugurated by Modi -- also collapsed Friday, reports said, with no one injured.
Delhi has been hit by heavy rains in recent days as the annual monsoon reached the Indian capital after a long stretch of heatwaves and punishingly high temperatures.
The downpours have brought the city to a standstill, with images shared by the city's police showing personnel clearing trees and helping residents stuck in waterlogged areas.