Some residents reporting being unable to contact relatives as water surged through towns and villages.
The Tonga Meteorological Services said the whole country, a small South Pacific nation of about 170 islands, was at risk.
As far away as Hawaii, about 5,000 kilometres away, flooding was reported in low-lying areas but no casualties were reported, while the Tokara and Amami island chains of Japan, which includes Okinawa, received a tsunami warning and were told to expect three-metre waves, Japanese broadcaster NHK said.
Authorities in the nearby island nations of Fiji and Samoa also issued warnings, telling people to avoid the shoreline owing to strong currents and dangerous waves.
Communications with Tonga was difficult. By Saturday evening, Tongans told New Zealand news service Stuff, that people had screamed for help as buildings were inundated.
Video posted to social media showed large waves washing ashore, swirling around homes and buildings.
Satellite images showed a huge explosion from the Hunga Tonga Hunga Ha’apai volcano, the latest in a series of spectacular eruptions.
Prof Shane Cronin, an expert on volcanos at the University of Auckland, told the BBC that the eruption could be one of the most powerful for 30 years.
Mere Taufa said she was in her house, getting ready for dinner, when the volcano erupted.
“It was massive. The ground shook, our house was shaking. It came in waves. My younger brother thought bombs were exploding nearby,” Ms Taufa told Stuff.
She said water filled their home minutes later and she saw the wall of a neighbouring house collapse.
“We just knew straight away it was a tsunami. Just water gushing into our home.”
New Zealand’s military said it was monitoring the situation and remained on standby, ready to assist if asked.
The Islands Business news site reported that a convoy of police and military troops had helped Tonga’s King Tupou VI to leave his palace near the shore. He was among many residents who fled for higher ground.
A video posted on Twitter on Tonga’s island cluster of Vava’u showed waves sweeping inland.
“Can literally hear the volcano eruption, sounds pretty violent,” Faka’iloatonga Taumoefolauhe wrote in his post.
He later wrote: “Raining ash and tiny pebbles, darkness blanketing the sky.”
Matangi Tonga, a news website, earlier reported that scientists observed massive explosions, thunder and lightning near the volcano after it started erupting early on Friday.
The site said satellite images showed a plume of ash, steam and gas rising up into the air to about 20 kilometres, and spanning 5km.
More than 2,300km away, in New Zealand, officials were warning of storm surges from the eruption.
Its National Emergency Management Agency said some parts of the country could expect “strong and unusual currents and unpredictable surges at the shore following a large volcanic eruption”.
On Saturday, the Pacific Tsunami Warning Centre said the threat to American Samoa appeared to have passed, although minor sea fluctuations could continue.
The volcano erupted about 64km north of the Tongan capital, Nuku’alofa. In late 2014 and early 2015, eruptions in the area created a new island and disrupted international air travel to the Pacific archipelago for several days.
Tonga is home to about 105,000 people.