After Mr Putin lit the cauldron, he said the "Olympic flame - symbol of the most important sports competition of this planet, symbol of peace and friendship - has arrived in Russia."
The flame then set out on a ceremonial loop through the Kremlin, with 11 torchbearers participating in a relay.
However, the flame, which was lit a week ago in Greece and flown to Moscow earlier on Sunday, went out.
The glitch occurred when torchbearer, former world swimming champion and record holder Shavarsh Karapetyan, ran through a long passageway leading into the Kremlin, which apparently created a wind tunnel, extinguishing the flame.
A security guard standing along the route, pulled out a lighter and re-lit the torch.
The torch relay will stay in Moscow for the next three days, with hundreds of athletes, cultural figures and others, including Prince Albert II of Monaco, taking part.
The flame will then begin its journey across Russia, travelling from the western exclave of Kaliningrad to the eastern-most point just across the Bering Strait from Alaska, before swinging back through the vast country to Sochi in time for the opening ceremony on February 7th.
For most of the 39,000-mile trip the flame will travel by plane, train, car and even reindeer-sleigh, safely encased inside a lantern.
One of the silver and red torches, unlit, will be carried into space in November for a brief visit to the International Space Station, and this same torch will be used to light the Olympic flame in Sochi.