Residents of Taiwan's fishing town of Yilan secured boats and homes as waves crashed on to shore.
“I will not worry. The typhoon won’t make landfall now,” fishing boat owner Wang Jian-chi told AP.
“The typhoon will move northward from the sea in the east of Taiwan. Its strength has also weakened. And there is no wind and waves in the fishing port at present. I don’t think it will affect us.”
The coastguard said precautions were being taken.
“We have issued a high surf warning. The wind and waves are very strong,” said coastguard officer Wang Hsing-chieh.
“We hope that beachgoers won’t come near the beach and embankment at this moment. We will also send coastguard personnel to patrol the port to warn beachgoers.”
Although the slow-moving typhoon has lost some of its ferocity since smashing into Guam last week, forecasters in the Philippines said Mawar remained dangerous with maximum sustained winds of 155kph and gusts of up to 190kph.
People in the Batanes province of the Philippines prepared for bad conditions, but were largely spared.
“I’m on the roof, but I’m not being blown away by the wind,” said Juliet Cataluna, a Batanes provincial official in the coastal town of Ivana.
“I wish we’ll really be spared from damages – our livelihood, our agricultural produce and our houses.”
After seeing earlier forecasts that Mawar would be stronger, residents in Ivana placed sandbags on their tin roofs and covered glass windows with wooden boards. Ms Cataluna wrapped her avocados with sack cloth so they would not be blown off trees.
Town leaders used motorbikes to deliver typhoon updates, she said, and fortunately only light rain and occasional wind gusts have hit Ivana.
The typhoon was offshore about 350km east of the Batanes capital Basco on Tuesday and is projected to shift north-east by Wednesday towards southern Japan.
Strong winds were still forecast for Taiwan, and authorities in the Philippines warned against complacency, saying the risks from dangerous tidal surges, flash floods, landslides and typhoon-enhanced monsoon rains remain until Mawar passes.
More than 3,400 villagers stayed in emergency shelters in northern provinces, flights to and from Batanes remained suspended, and classes have not resumed in more than 250 cities and towns in the north, the Office of Civil Defence said.
Winds lashed nearby Cagayan province on Monday, causing an unoccupied wharf warehouse to collapse and prompting more villagers to move to evacuation centres.
Mawar tore through Guam last week as the strongest typhoon to hit the US Pacific territory in over two decades, flipping cars, tearing off roofs and knocking out power.