Three suspects have been detained and two others are on the run, the source said.
Two people were inside the store at the time and alerted police. "Passers-by took refuge in the hotel".
Terrified guests at the hotel were left fearing for their lives as the attackers destroyed display cases inside a posh jewellery store at the hotel before making off with the stash.
No injuries were reported in the incident, although some reports on Twitter said that the robbers burst into the Hemingway bar in the hotel and fired their weapons.
Interior minister Gerard Collomb praised officers' "professionalism", saying they had "done our police force credit".
Another put the figure at 4.5 million euros ($NZ7.5 million), but said that a bag had been recovered possibly containing some of the loot.
Place Vendome, where the Ritz is based, is also home to the Ministry of Justice, and several luxury brands, including Chanel. Three men were charged in connection with that heist.
Kim Kardashian West lost millions of dollars worth of jewellery when she was robbed at gunpoint in a Paris apartment in October 2016.
In more recent history however, the Ritz has become known as the place where Britain's Princess Diana spent her last hours before a auto accident in a tunnel in the French capital while being pursued by paparazzi.