The agency said at least 637 people have been injured, and about 15,000 others have been forced to evacuate.
Rescue teams were searching for more than a dozen patients and staff trapped in the rubble of a flattened Mamuju hospital. "Please help me, it's hurt", the girl told rescuers, who replied that they desperately wanted to help her.
The quake damaged part of a hospital and patients were moved to an emergency tent outside.
A video circulating on social media showed rescue workers checking on two sisters who were trapped under rubble.
Another video showed a father crying, asking for help to save his children buried under their toppled house. "My children there. they are trapped inside, please help", he cried in panic.
The quake was centered 36 kilometers (22 miles) south of West Sulawesi province's Mamuju district, at a depth of 18 kilometers (11 miles), the U.S. Geological Survey said. The local airport in Mamuju had also been damaged, authorities said.
A man grieves over the body of a victim in Mamuju city on Friday, aftwr a 6.2-magnitude quake rocked Indonesia's Sulawesi island.
People look at the damaged governor of West Sulawesi province's office following an quake in Mamuju. He said his team was working to save 20 people trapped in eight buildings, including in the governor's office, a hospital and hotels.
In the video, the rescuers said an excavator was needed to save them.
Relatives wailed as they watched rescuers pull a body of a loved one from a damaged home in devastated Mamuju.
The natural disaster also triggered power outage and caused three landslides along the main road connecting Majene and Mamuju.
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In a message to his Indonesian counterpart Retno LP Marsudi, Dr Momen expressed his heartfelt sympathies to the Indonesian government, people of the country, and the members of the bereaved families.
"The hospital is flattened - it collapsed", said Arianto from the rescue agency in Mamuju city, who goes by one name.
Puruhito is already leading more than 4,100 rescue personnel in a separate massive search operation for victims of the crash of a Sriwijaya Air jet into the Java Sea last Saturday.
Thousands are reported to have left their homes, fleeing to safety.
"Some of them are going to the higher place to avoid tsunami, although we have a confirmation that we have no tsunami after this big natural disaster", he said.
National Disaster Mitigation Agency spokesman Raditya Jati said three people had been killed and 24 injured.
Hours earlier on Thursday, a 5.9-magnitude quake struck in the same district damaging several houses, but causing no apparent casualties.
The meteorological agency warned residents that the area could be hit by strong aftershocks and to avoid the beachfront in case of a tsunami.
Indonesia has a history of devastating earthquakes and tsunamis with more than 2,000 killed in a 2018 Sulawesi quake. "Numerous dead are buried under rubble".
A 9.1-magnitude quake off the north of Sumatra island triggered a tsunami on Boxing Day in 2004 that lashed coastal areas of Indonesia, Sri Lanka, India, Thailand and nine other nations, killing more than 230,000 people.