There were no immediate reports of injuries or the extent of the damage as communications with the small Pacific nation remain problematic.
Video posted to social media on Saturday showed large waves washing ashore in coastal areas, swirling around homes and buildings.
A spokesperson for the Australian government said initial assessments are under way and the Department of Foreign Affairs is working to ensure Australians in Tonga are safe and accounted for.
"Tonga is part of our Pacific family and our thoughts are with the entire community dealing with the impact of the volcanic eruption and tsunami," the spokesperson said in a statement.
"The Prime Minister and Minister for Foreign Affairs are monitoring the situation and Australia stands ready to provide support to Tonga if requested."
New Zealand's military said it was monitoring the situation and remained on standby, ready to assist if asked.
Satellite images showed a huge eruption. The Tonga Meteorological Services said a tsunami warning was in effect for all of Tonga, and data from the Pacific Tsunami Warning Centre showed waves of 80cm had been detected.
Authorities in nearby Fiji and Samoa also issued warnings, telling people to avoid the shoreline due to strong currents and dangerous waves.
The Japan Meteorological Agency said there may be a slight swelling of the water along the Japanese coasts, but it is not expected to cause any damage.
The Islands Business news site reported that a convoy of police and military troops evacuated Tonga's King Tupou VI from his palace near the shore. He was among the many residents who headed for higher ground.
The explosion of the Hunga Tonga-Hunga Ha'apai volcano was the latest in a series of spectacular eruptions.
A Twitter user identified as Dr Faka'iloatonga Taumoefolau posted video showing waves crashing ashore.
"Can literally hear the volcano eruption, sounds pretty violent," he wrote, adding in a later post: "Raining ash and tiny pebbles, darkness blanketing the sky."
Earlier, the Matangi Tonga news site reported that scientists observed massive explosions, thunder and lightning near the volcano after it started erupting early Friday.
The site said satellite images showed a 5km-wide plume of ash, steam and gas rising up into the air to about 20km.
The volcano is located about 64km north of the capital, Nuku'alofa.
Back in late 2014 and early 2015, a series of eruptions in the area created a small new island and disrupted international air travel to the Pacific archipelago for several days.
Tonga is home to about 105,000 people.
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