The raid on the store near the casino building in the centre of the principality was carried out in broad daylight on Saturday.
Police sealed off all roads leading out of Monaco immediately after the raid and ordered people to remain inside the casino and nearby shops while they searched for the suspects.
One man was arrested shortly after the raid and a gun was recovered and another two men were picked up in Monaco later Saturday, the source said.
A fourth man was arrested in France.
Armed robberies are rare in Monaco, one of Europe's playgrounds for the rich, although in 2012 four Lithuanian men stole a collection of around 50 watches worth around €200,000 from a jewellery store.
The men were later arrested and at trial were sentenced to prison sentences of between five and 13 years.
A Cartier store was also the target in 2015 when a masked man carrying a machine pistol burst into a store in the French Riviera resort of Cannes during the annual film festival.
He let two accomplices enter the shop and they stole watches and jewels with a value of €17.5 million ($18.9 million).