The US warned of strong currents and waves, and coastal flooding.
The huge underwater volcanic eruption caused waves of more than a metre to crash into Tonga.
The eruption of the Hunga Tonga-Hunga Haʻapai volcano was heard across the South Pacific, and eventually as far away as the US.
A tsunami warning has been issued for parts of Australia's east coast and Tasmania, and the north and east coast areas of New Zealand's North Island.
Many parts of Tonga, whose capital is just 65km south of the eruption, are covered in ash and are experiencing a near-total blackout of power, phone lines and internet services. The extent of any injuries or damage is still unclear.
Social media footage showed water washing through a church and several homes, and witnesses said ash was falling over the capital, Nuku'alofa.
Videos on social media showed traffic jams as people tried to flee low-lying areas by car.
The plumes of gas, smoke and ash pouring from the volcano reached 20km into the sky, Tonga Geological Services said.
In Japan, the 1.2m tsunami was recorded in the Kominato district of Amami-Oshima Island in Kagoshima Prefecture at 23:55 (14:55 GMT) on Saturday.
So violent was the initial eight-minute eruption that it could be heard as "loud thunder sounds" in Fiji, more than 800km away, according to officials in the capital, Suva.
The Fijian government issued a tsunami advisory and opened evacuation centres for people in coastal areas. Vanuatu, another island nation in the Pacific, issued a similar warning.
Professor Shane Cronin, a volcanologist at the University of Auckland, said the eruption was one of the biggest in Tonga in the past 30 years.
"This is a pretty big event - it's one of the more significant eruptions of the last decade at least," he told the BBC.
"The most remarkable thing about it is how rapidly and violently it's spread. This one was larger, a much wider lateral spread, much more ash was produced. I expect there to be many centimetres of ash that have been deposited on Tonga."
The Australian and New Zealand governments have said they were monitoring the situation and ready to provide support on request.