There were no immediate reports of injuries or the extent of the damage as communications with the small nation remained problematic.
Video posted to social media showed large waves washing ashore in coastal areas, swirling around homes and buildings.
New Zealand's military said it was monitoring the situation and remained on standby to assist if asked.
Satellite images showed a huge eruption. Tonga Meteorological Services said a tsunami warning was in effect for all of Tonga.
Authorities in the nearby island nations of Fiji and Samoa also issued warnings, telling people to avoid the shoreline due to strong currents and dangerous waves.
The Islands Business news site reported that a convoy of police and military troops had transported Tonga's King Tupou VI from his palace near the shore as residents 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 on Friday.
The site said satellite images showed a three mile-wide plume of ash, steam and gas rising 12 miles into the air.
More than 1,400 miles away in New Zealand, officials warned of storm surges from the eruption.
The National Emergency Management Agency said some parts could expect "strong and unusual currents and unpredictable surges at the shore following a large volcanic eruption".
Late 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 is about 40 miles north of the Tongan capital Nuku'alofa. 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.