Accusations have been centred on the country's meteorology and geophysics agency (BMKG), with some people accusing the agency of turning off the tsunami warning too early.
An alert was sounded warning of waves potentially as high as three metres shortly after 6pm on Friday, which was cancelled at 6.36 pm before the tsunami hit.
However, the agency said the alert was lifted only after the tsunami hit.
In a statement, the head of the agency said Dwikorita Karnawati said the allegations were "not correct."
"Our work is based on computer system/artificial intelligence. The warning system was lifted with the approval of the 28 other countries along the Indian ocean," Mr Karnawati said.
Tsunami warning failed on 'last mile'
It comes as the German research centre that developed a warning system used by Indonesia said tsunami warnings to the local population in Sulawesi island failed on the "last mile", causing many to be surprised by waves as high as six metres.
"The problem was the communication between local authorities and people, for example on the beach, such as in Sulawesi," Joern Lauterjung, Director Geoservices at GFZ, told Reuters TV.
Germany provided a warning system developed by GFZ to Indonesia after a devastating tsunami killed 226,000 people in 2004.
Lauterjung said that system worked as planned, predicting waves up to three metres northwest of Sulawesi.
"If you look at the entire warning chain from the creation of a warning signal up to the last mile, as we call it, up to the local population in danger, there was a problem there," he said.
"For example, it appears sirens did not work and there were no warnings via loudspeaker vans from police to the local population," he added.
Indonesia scrambled on Monday to get help into quake-hit Sulawesi island as survivors streamed away from their ruined homes and accounts of devastation filtered out of remote areas, including the death of 34 children at a Christian camp.
Dozens of people were reported to be trapped in the rubble of several hotels and a mall in the small city of Palu, 1,500 km northeast of Jakarta. Hundreds more were feared buried in landslides that engulfed villages.
Of particular concern is Donggala, a region of 300,000 people north of Palu and close to the epicenter of the quake, and two other districts, where communication had been cut off.
The four districts have a combined population of about 1.4 million.
In response, President Joko Widodo opened the door to the dozens of international aid agencies and NGOs who are lined up to provide life-saving assistance.
Officials fear the toll will rise steeply in the coming days and are preparing for the worst, declaring a 14-day state of emergency.
The United Nations Office for the Coordination of Humanitarian Affairs warned that there were some 46,000 children and 14,000 elderly Indonesians among those in dire need -- many in areas that aren't the focus of government recovery efforts.
One woman was recovered alive from ruins overnight in the Palu neighborhood of Balaroa, where about 1,700 houses were swallowed up when the earthquake caused soil to liquefy, the national rescue agency said.
“We don’t know how many victims could be buried there, it’s estimated hundreds,” said Sutopo Purwo Nugroho, spokesman for the National Disaster Mitigation Agency.
All but 23 of the confirmed deaths were in Palu, a city of about 380,000 people, where workers were preparing a mass grave to bury the dead as soon as they were identified.
Nearly four days after the quake, the extent of the disaster was not known with authorities bracing for the toll to climb - perhaps into the thousands - as connections with remote areas up and down the coast are restored.
Aid worker Lian Gogali, who had reached Donggala district by motorcycle, said hundreds of people facing a lack of food and medicine were trying to get out, but evacuation teams had yet to arrive and roads were blocked.
“It’s devastating,” she told Reuters by text.
Indonesian Red Cross spokeswoman Aulia Arriani said a church in an area of Sigi, south of Palu, had been engulfed in mud and debris. Officials said the area suffered liquefaction, when the shock of the quake temporarily destabilizes the soil.
“My volunteers found 34 bodies ... children who had been doing a bible camp,” Arriani said.
Sulawesi is one of the earthquake-prone archipelago nation’s five main islands and sits astride fault lines. Numerous aftershocks have rattled the region.
Pictures showed expanses of splintered wood, washed-up cars and trees mashed together, with rooftops and roads split asunder. Access to many areas is being hampered by damaged roads, landslides and collapsed bridges.
A Reuters witness said queues at petrol stations on the approaches to Palu stretched for miles. Convoys carrying food, water and fuel awaited police escorts to prevent pilfering before heading toward the city while residents streamed out.
The state energy company said it was airlifting in 4,000 liters of fuel, while Indonesia’s logistics agency said it would send hundreds of tonnes of rice. The government has allocated 560 billion rupiah ($37.58 million) for the recovery.
The government has played down worries about looting though witnesses have seen incidents.
Chief security minister Wiranto said more than 2,800 troops had been deployed and plans were in place to send in a further 2,000 police.
The government would accept offers of help from 18 countries and it had also commandeered 20 excavators from mines and plantations to help with a shortage of equipment to dig through wreckage and clear blocked roads, he said.
Nearly 60,000 people were displaced, many terrified by powerful aftershocks, and they needed tents, water and sanitary facilities, while the power utility was working to restore electricity, he said.
Commercial flights have yet to resume but military aircraft were taking people out of Palu. About 3,000 people thronged the small airport hoping to get out and officers struggled to keep order.
“I’d get a plane anywhere. I’ve been waiting for two days. Haven’t eaten, barely had a drink,” said 44-year-old food vendor Wiwid.
Indonesia is all too familiar with earthquakes and tsunamis. A quake in 2004 triggered a tsunami across the Indian Ocean that killed 226,000 people in 13 countries, including more than 120,000 in Indonesia.
Palu sits astride the Palu-Koro fault, which runs north-south along the edge of Palu Bay. Geologists estimate segments of the fault have a slip that is among the highest in Indonesia, at 4 cm (1.6 inches) a year, exposing the area to a higher risk of quakes.
Questions are sure to be asked why warning systems set up after the 2004 disaster appear to have failed.
Disaster agency spokesman Nugroho told reporters on Sunday none of Indonesia’s tsunami buoys, one device used to detect waves, had been operating since 2012. He blamed a lack of funds.
The meteorological and geophysics agency BMKG issued a tsunami warning after the quake but lifted it 34 minutes later, drawing criticism it had been too hasty.
However, officials estimated the waves had hit while the warning was in force.