The epicentre of the quake was about 10 kilometres south of Bodrum, a magnet for holidaymakers, and 16 kilometres east of Kos, the US Geological Survey said.
"(There was) this loud rumbling noise and we all ran out... My four friends fell to the ground due to the place shaking so much," 18-year-old Briton Harriet Longley told AFP from Kos.
"We all crawled out and managed to get down the stairs where the other guests were screaming.
Television footage showed gutted stone buildings and island streets filled with rubble.
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The quake, followed by scores of aftershocks, also toppled the minaret of an 18th century Ottoman mosque and cracked the dock in the port of Kos, which has been shut down.
A small tsunami sent fishing boats crashing into Kos harbour and damaged cars in the resort of Gumbet outside Bodrum.
Police said a 22-year-old Swede and a 39-year-old Turk died in a mediaeval-era area full of cafes and nightclubs in Kos. Another man from Sweden lost his lower leg, and doctors are struggling to save his other leg, officials said.
About 120 people were hurt in Kos and nearly 360 in Bodrum, many of them after jumping out of windows, officials and media reports said.
Ministers returning from a swift inspection of Kos insisted overall damage was minimal and Greek Prime Minister Alexis Tsipras declared the island was "coming back to normality".
"Kos is safe, there is no substantial problem," deputy shipping minister Nektarios Santorinios told a news conference.
The injured were found on the street in the Kos club area, crushed either by the collapsed wall of a bar or by stones that fell from old houses nearby.
With Kos airport also temporarily shut for safety tests, hundreds had to queue as several early morning flights were cancelled or delayed.
The airport is now operating at full capacity, German handling company Fraport said.
"There is no panic, those leaving are on scheduled flights," an airport source told AFP.
"The airport is operational and the roads are in good shape... There is no major damage to infrastructure or buildings," said government spokesman Dimitris Tzanakopoulos.
Kos attracts young tourists and during peak season has a capacity of around 100,000 beds, a local police official said, adding that more than 85 per cent were currently booked.
Until then, ferry traffic is being rerouted to the neighbouring island of Kalymnos.
A ferry has also been sent to evacuate 200 Turkish nationals from Kos.
The quake struck at 1:31 am Friday (local time) and was also felt on the Greek island of Rhodes.
An AFP journalist staying in Theologos, about 30 kilometres from Rhodes town, said the hotel "rocked like a boat and I thought it was going to collapse".
"We were very surprised. We were scared and we immediately went outside," 38-year-old Teddy Dijoux, who was holidaying with his family, told AFP.
The Adliye mosque in central Bodrum also suffered some damage.
"The biggest problem at the moment are electricity cuts in certain areas," Bodrum mayor Mehmet Kocadon told Turkey's NTV television.
He said damage was light and there were no reports of deaths.
Turkey and Greece sit on significant fault lines and have regularly been hit by earthquakes in recent years.
This year alone, Turkey's western Aegean coast was hit by several significant earthquakes.
In June, a 6.3-magnitude earthquake gutted a village on the Greek island of Lesbos, killing a woman and leaving more than 15 injured.
On August 17, 1999, a huge earthquake measuring more than 7.0 magnitude near the Turkish city of Izmit devastated vast areas in the densely populated northwest, notably around Istanbul, killing over 17,000 people.