The 6.4 magnitude quake hit the central island of Lombok just before 07:00 local time (00:00 GMT) on Sunday.
The island attracts tourists from around the world due to its beaches and hiking trails, and is located about 40km (25 miles) east of Bali.
More than 160 people are injured and thousands of homes are damaged, officials say.
A Malaysian tourist who was on a hiking trip to Mount Rinjani is among those killed.
The US Geological Survey said the epicentre of the earthquake was 50km (31 miles) north-east of the city of Mataram, in northern Lombok.
It was followed by more than 60 smaller earthquakes, with the largest recorded at a magnitude of 5.7.