The half-tonne Vikram-S rocket launched before midday local time and travelled in an arc, live footage from the Indian Space Research Organisation (Isro) showed.
It splashed down into the sea six minutes later, according to the agency.
The rocket, developed by local start-up Skyroot Aerospace, reached a peak altitude of 90km, below the internationally recognised 100km Karman line that separates Earth from outer space.
"It is indeed a new beginning, a new dawn ... in the journey of India's space programme," Science Minister Jitendra Singh said after the launch to a crowd of cheering technicians at the Isro launch facility on the southern island of Sriharikota.
The single-stage, solid-fuel rocket was built with "carbon composite structures and 3D-printed components", the government said on Thursday ahead of the first Vikram-S mission, named Prarambh (Start in English).
India has been bolstering its space programme in recent years, including a crewed mission with Russian backing scheduled for 2023 or 2024.
Skyroot, which was started by Pawan Chandana and Bharath Daka, has set a target of cutting development costs by up to 90 per cent compared with existing platforms to launch small satellites.
It expects to achieve those cost savings by using a rocket architecture that can be assembled in less than 72 hours with composite materials. It plans launches capable of delivering satellites starting next year.
"Innovation and cost efficiency should be the two drivers for the industry. Cost efficiency has already been achieved, and now we should look at cutting-edge technology," Mr Chandana said.
India's capabilities and ambitions have grown, highlighted by the success of its rockets and missions beyond Earth.
In 2014, India became the first Asian nation to reach Mars with its Mangalyaan orbiter. Hailed for its low cost, that mission put India in a small club including the United States, Russia and the European Union.
In 2019, Prime Minister Narendra Modi hailed India as a "space superpower" after it shot down a low-orbiting satellite, a move prompting criticism for the amount of "space junk" it created.
India is also working to boost its 2 per cent share of the global commercial space market.
In October, Isro's heaviest rocket yet successfully put 36 broadband satellites into low-Earth orbit.