The painting, entitled "Ballerina adjusting her slipper", was taken on Monday from the home of the 70-year-old Greek Cypriot collector in the island's second city, Limassol.
It was not insured, according to police.
Art theft is rare in Cyprus and it is believed to be one of the most valuable paintings ever stolen on the Mediterranean island.
The thieves also made off with seven gold watches and three gold opera glasses, worth a total of 157,000 euros.
Police said they had arrested a 44-year-old Greek Cypriot in connection with the case and were seeking two other suspects, a South African and a Russian. It is believed the trio knew the victim.
The French artist Degas is regarded as one of the founders of Impressionism.
More than half of his paintings, sculpures and drawings depict dancers.
The painting which was stolen is one of a number of studies made by Degas between 1873 and 1874 of dancers adjusting their shoes, shown in different poses and from different angles.