Scientists at the mission control centre in Sriharikota, off the Bay of Bengal, burst into applause as the rocket lifted off in clear weather as scheduled at 2.43pm local time (10.13 BST) Monday.
Chandrayaan, the Sanskrit word for mooncraft, is designed to land on the lunar south pole in September and send a rover to explore water deposits that were confirmed by a previous mission which orbited the moon.
India’s first moon mission orbited the moon in 2008 and helped confirm the presence of water.
India plans to send its first manned spaceflight by 2022.
Prime minister Narendra Modi said India’s lunar programme will get a substantial boost, writing on Twitter that the country’s existing knowledge of the moon “will be significantly enhanced”.
K. Sivan, head of India’s space agency, said at a news conference that the successful launch of the spacecraft was the “beginning of India’s historic journey” to the moon.
The launch a week ago was called off less than an hour before lift-off due to a “technical snag”.
Media reports said the launch was aborted after scientists identified a leak while filling helium in the rocket’s cryogenic engine.
The space agency neither confirmed nor denied the reports, saying instead that the problem had been identified and corrected.
The spacecraft is carrying an orbiter, a lander and a rover that will move around on the lunar surface for 14 earth days.
It will take about 47 days to travel before landing on the moon in September.
India put a satellite into orbit around Mars in the nation’s first interplanetary mission in 2013 and 2014.