The robotic vehicle rolled down a ramp lowered by the lander and on to the volcanic plain known as Sinus Iridum.
Earlier on Saturday, the landing module containing the rover fired its thrusters to perform the first soft landing on the Moon since 1976.
The touchdown in the Moon’s northern hemisphere marks the latest step in China’s ambitious space programme.
The lander will operate there for a year, while the rover is expected to work for some three months.
“I was lucky enough to see a prototype rover in Shanghai a few years ago – it’s a wonderful technological achievement to have landed,” Prof Andrew Coates, from UCL’s Mullard Space Science Laboratory, told BBC News.
Chang’e-3 is the third unmanned rover mission to touch down on the lunar surface, and the first to go there in more than 40 years. The last was an 840kg (1,900lb) Soviet vehicle known as Lunokhod-2, which was kept warm by polonium-210.
But the six-wheeled Chinese vehicle carries a more sophisticated payload, including ground-penetrating radar which will gather measurements of the lunar soil and crust.
The 120kg (260lb) Jade Rabbit rover can reportedly climb slopes of up to 30 degrees and travel at 200m (660ft) per hour.
Its name – chosen in an online poll of 3.4 million voters – derives from an ancient Chinese myth about a rabbit living on the moon as the pet of the lunar goddess Chang’e.
The rover and lander are powered by solar panels but some sources suggest they also carry radioisotope heating units (RHUs), containing plutonium-238 to keep them warm during the cold lunar night.
Reports suggest the lander and rover will photograph each other at some point on Sunday.
According to Chinese space scientists, the mission is designed to test new technologies, gather scientific data and build intellectual expertise. It will also scout valuable mineral resources that could one day be mined.
“China’s lunar programme is an important component of mankind’s activities to explore [the] peaceful use of space,” Sun Huixian, a space engineer with the Chinese lunar programme, told Xinhua.
After this, a mission to bring samples of lunar soil back to Earth is planned for 2017. And this may set the stage for further robotic missions, and – perhaps – a crewed lunar mission in the 2020s.
Dean Cheng, a senior research fellow at the Heritage Foundation, a conservative think-tank in Washington DC, said China’s space programme was a good fit with China’s concept of “comprehensive national power”. This might be described as a measure of a state’s all-round capabilities.
But he said that China did not see itself as being in a “space race” with anyone else. “I’m comparing it specifically to how the US and the Soviets were behaving in the late 1950s and 1960s when you had space launches almost every month.
“Look at how often the Chinese do manned missions – it’s almost every other year.”
Instead, the country is methodically and patiently building up the key elements needed for an advanced space programme – from launchers to manned missions in Earth orbit to unmanned planetary craft – and it is investing heavily.
Mr Cheng added: “China is saying: ‘We are doing something that only two other countries have done before – the US and the Soviet Union.”
The landing site of Sinus Iridum (Latin for Bay of Rainbows) is a flat volcanic plain, part of a larger feature known as Mare Imbrium that forms the right eye of the “Man in the Moon”.