A day before sweeping tariffs came into effect on dozens of countries, the US president said Washington would also be placing a "100 percent" tariff on chips and semiconductors but he did not offer a timetable.
However, he said "the good news for companies like Apple is, if you're building in the United States, or have committed to build... in the United States, there will be no charge".
Stock gains were led Thursday by Taiwan's giant TSMC, which surged almost five percent, with the island's National Development Council chief Liu Chin-ching saying the firm was in the clear.
"Because Taiwan's main exporter is TSMC, which has factories in the United States, TSMC is exempt," he told a briefing in parliament.
TSMC, which is ramping up manufacturing in Arizona, has pledged to invest as much as $165 billion in the United States.
Seoul-listed Samsung, which is also pumping billions into the world's number one economy, rose more than two percent while South Korean rival SK hynix was up more than one percent.
Apple-linked firms were helped after the US giant said it would invest an additional $100 billion in the United States, taking its total pledge to $600 billion over the next four years.
However, Japanese trade Tokyo Electron, a major producer of chipmaking equipment, plunged more than two percent, while chipmaker Renesas sank 3.8 percent.
Precision tools maker Disco Corporation gave up 1.8 percent.
Sony soared 4.1 percent after the PlayStation-maker raised its annual profit forecasts, citing strong performance in its key gaming business and a smaller-than-expected negative impact of US trade tariffs.
- Tariff talks -
Analysts said that while the chip threat was steep, there was optimism the final level would be lower.
"The figure fits Trump's approach of 'open high, negotiate down' and the final figure could be similar to reciprocal tariffs to limit inflation in consumer goods -- given that many have chips," said Morningstar's Phelix Lee.
Trump's remarks came hours before his sweeping "reciprocal" tariffs kicked in Thursday against trading partners, and after he doubled his levy on India to 50 percent over its purchase of Russian oil.
Fifty percent tolls on Brazilian goods came into place Wednesday.
Asian markets extended their recent run-up following a strong day on Wall Street, where Apple jumped more than five percent and Amazon piled on four percent.
Tokyo, Hong Kong, Singapore, Seoul, Bangkok, Jakarta and Wellington were all in the green, with Taipei leading the way thanks to the surge in TSMC.
Shanghai finished on a positive note after data showed Chinese exports rose more than expected, with a surge in shipments to the European Union and Southeast Asian nations offsetting a more than 20 percent plunge in those to the United States.
Imports also climbed, providing a boost to efforts to kick-start the Chinese economy.
Mumbai fell, along with Sydney and Manila as well as London. Paris and Frankfurt edged up.
Traders had already been on a buying streak on optimism the Fed will cut rates after data last week showing US jobs creation cratered in May, June and July, signalling the economy was weakening. US futures rose.
Oil prices also rose after Trump threatened penalties on other countries that "directly or indirectly" import Russian oil, after imposing his extra toll on India.
Traders are keeping tabs on developments regarding Moscow and its war in Ukraine after the US president said he could meet with Vladimir Putin "very soon". That followed what he called highly productive talks between his special envoy and the Russian leader.
- Key figures at around 0810 GMT -
Tokyo - Nikkei 225: UP 0.7 percent at 41,059.15 (close)
Hong Kong - Hang Seng Index: UP 0.7 percent at 25,081.63 (close)
Shanghai - Composite: UP 0.2 percent at 3,639.67 (close)
London - FTSE 100: DOWN 0.4 percent at 9,129.05
Euro/dollar: UP at $1.1692 from $1.1659 on Wednesday
Pound/dollar: UP at $1.3376 from $1.3358
Dollar/yen: DOWN at 146.89 yen from 147.38 yen
Euro/pound: UP at 87.41 pence from 87.23 pence
West Texas Intermediate: UP 0.7 percent at $64.79 per barrel
Brent North Sea Crude: UP 0.6 percent at $67.28 per barrel
New York - Dow: UP 0.2 percent at 44,193.12 (close)
dan/sco
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