Chandrayaan-2, which means 'moon vehicle' in Sanskrit, entered the orbit around midnight Eastern Time in a complicated insertion maneuver that took 29 minutes.
'Our hearts almost stopped today till it completed its job,' Kailasavadivoo Sivan, chairman of the Indian Space Research Organization (ISRO), told reporters at a news conference.
'But still the landing is the terrifying moment... because that is a phase that we are doing for the first time,' he added.
If the spacecraft lands successfully, India will become only the fourth country to make a soft landing on the moon, after the United States, the former Soviet Union and China.
India's Prime Minister Narendra Modi congratulated the ISRO on Tuesday, tweeting: 'Best wishes for its successful culmination.'
The spacecraft will next perform a series of orbit maneuvers, positioning itself into a circular orbit passing over the lunar poles at a distance of about 62 miles (100 km) from the Moon's surface.
From there, the lander -- named Vikram, after the pioneer of the Indian space program Vikram Sarabhai -- will separate from the main vessel and is planned to gently touch down on the moon's surface near its south pole.
The ISRO said Tuesday it expects the landing to happen on September 7.
A robotic rover named Pragyan (meaning 'wisdom') will then deploy and spend one lunar day, or 14 Earth days, collecting mineral and chemical samples from the moon's surface for remote scientific analysis.
Over the next year, the orbiter will map the lunar surface and study the outer atmosphere of the moon.
The Chandrayaan-2 weighs 3.8 tons and carries 13 payloads. It has three elements -- lunar orbiter, lander and rover -- all developed by the ISRO.
It took off from the Satish Dhawan Space Center at Sriharikota in Andhra Pradesh state on July 22.
The launch was originally scheduled for July 15, but was abruptly called off just 56 minutes before liftoff due to a 'technical snag.'