The half-tonne Vikram-S rocket took off before 12:00 p.m. (0700 GMT), before completing an arc trajectory, as shown by images from the Indian Space Research Organization (ISRO).
Six minutes later, she sank safely into the sea, according to the agency.
This rocket reached an altitude of 90 km above sea level, below the Karman line which marks the boundary of space according to international convention.
"It is indeed a new beginning, a new dawn (...) for the Indian space project", declared the Indian Minister of Sciences, Jitendra Singh, at the end of this launch, in front of the enthusiastic technicians present on the ISRO launch site on the island of Sriharikota, in the state of Andhra Pradesh (south).
The single-stage rocket, powered by solid propellant, was developed by the Indian start-up Skyroot Aerospace.
It was built with "composite carbon structures and 3D printed components," the government said in a statement on Thursday.
India has boosted its space program in recent years, planning a manned mission with support from Russia and France scheduled for 2023 or 2024.
In 2019, Mr Modi welcomed India joining the club of "space superpowers" after shooting down a satellite in low orbit with a missile.
But his country had drawn criticism because of the quantities of "space waste" thus generated.
In October, ISRO's heaviest rocket successfully placed 36 broadband satellites into low orbit.
According to experts, India manages to keep costs low by replicating and adapting existing space technology for its own purposes, not least thanks to its large number of highly skilled engineers who are paid far less than their foreign counterparts.
© 2022 AFP