"Our scientific community has once again done the country proud and the entire nation salutes them," Mr Singh said in a message.
The launch was greeted with applause by scientists gathered at the site.
The chief of the Indian Space Research Organisation (ISRO), Madhavan Nair, said it was a "historic moment" for the country.
"Today what we have charted is a remarkable journey for an Indian spacecraft to go to the moon and try to unravel the mysteries of the Earth's closest celestial body and its only natural satellite," Mr Nair said.
The BBC's Sanjoy Majumder in Delhi says there has been a lot of excitement about the event, which was broadcast live on national TV.
An Indian-built launcher carrying the one-and-a-half-tonne satellite blasted off from Satish Dhawan Space Centre in Sriharikota, an island off the coast of Andhra Pradesh, at about 0620 local time (0050 GMT).
One key objective will be to search for surface or sub-surface water-ice on the Moon, especially at the poles.
Another will be to detect Helium 3, an isotope which is rare on Earth, but is sought to power nuclear fusion and could be a valuable source of energy in future.
Powered by a single solar panel generating about 700 Watts, the Indian Space Research Organisation (ISRO) probe carries five Indian-built instruments and six constructed in other countries, including the US, Britain and Germany.
The mission is expected to cost 3.8bn rupees (£45m; $78m), considerably less than Japanese and Chinese probes sent to the Moon last year.
The Indian experiments include a 30kg probe that will be released from the mothership to slam into the lunar surface. The Moon Impact Probe (MIP) will record video footage on the way down and measure the composition of the Moon's tenuous atmosphere.
It will also drop the Indian flag on the surface of the Moon. The country's tricolour is painted on the side of the probe and, if successful, India will become the fourth country after the US, Russia and Japan to place its national flag on the lunar surface.
Professor David Southwood, director of science and robotic exploration at the European Space Agency congratulated India's space agency on the launch.
He added: "While the exploration of space calls for new challenges to be overcome, joining forces is becoming more and more a key to future successes."
Barry Kellett, project scientist on the C1XS instrument, which was built at the Rutherford-Appleton Laboratory in the UK, said: "Chandrayaan has a very competitive set of instruments... it will certainly do good science."
C1XS will map the abundance of different elements in the lunar crust to help answer key questions about the origin and evolution of Earth's only natural satellite.
Researchers say the relative abundances of magnesium and iron in lunar rocks could help confirm whether the Moon was covered by a molten, magma ocean early on in its history.
"The iron should have sunk [in the magma ocean], whereas the magnesium should have floated," Mr Kellett told BBC News.
"The ratio of magnesium to iron for the whole Moon tells you to what extent the Moon melted and what it did after it formed."
The instrument will look for more unusual elements on the Moon's surface, such as titanium. This metallic element has been found in lunar meteorites, but scientists know little about its distribution in the lunar crust.
Chandrayaan (the Sanskrit word for "moon craft") will also investigate the differences between the Moon's near side and its far side. The far side is both more heavily cratered and different in composition to the one facing Earth.
The spacecraft will take about eight days the reach the Moon. During its lunar encounter, Chandrayaan fires its engine to slow its velocity - allowing it to be captured by the Moon's gravity.
Chandrayaan will then slip into a near-circular orbit at an altitude of 1,000km. After a number of health checks, the probe will drop its altitude until it is orbiting just 100km above the lunar surface.
India, China, Japan and South Korea all have eyes on a share of the commercial satellite launch business and see their space programmes as an important symbol of international stature and economic development.
Last month, China became only the third country in the world to independently carry out a spacewalk.
But the Indian government's space efforts have not been welcomed by all.
Some critics regard the space programme as a waste of resources in a country where millions still lack basic services.
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