The gains in Asia came after major indexes stateside closed the Monday session significantly higher as investors looked past geopolitical tensions that had clouded markets for most of last week. The Dow Jones industrial average gained 0.62 percent, or 135.39 points, to end at 21,993.71. The CBOE Volatility Index (VIX), widely considered the best gauge of fear in the market, traded near 12.3, down about 20 percent.
The dollar edged up against a basket of currencies following overnight gains, with the dollar index at 93.717 at 3:08 p.m. HK/SIN. The greenback also firmed against the yen, with the U.S. currency last fetching 110.41 yen, climbing back to the 110 handle seen before geopolitical tensions ramped up last week.
"Traders cited the easing of tensions with North Korea as the war of words appears to have abated, but the move appeared to be a classic short covering rally as the dollar rebounded after several days of heavy selling," said BK Asset Management Managing Director of FX Strategy Boris Schlossberg in a Monday evening note.
Tensions appeared to de-escalate further, as North Korean leader Kim Jong Un told state media Tuesday that he would observe actions taken by the U.S. before making further decisions, Reuters reported. North Korea had said last week it was considering plans to strike Guam.
"Historically, shares have been adversely affected initially on the uncertainty of wars — or threatened wars — with a potential significant economic impact, but have tended to rally well before the conflict is over," said AMP Capital Head of Investment Strategy and Chief Economist Shane Oliver in a note.
With geopolitical risk easing, investors could turn their focus back to monetary policy, IG Market Strategist Jingyi Pan told CNBC. "The rest of the week holds (the release of) Fed minutes ahead of the Jackson Hole meeting next week, and those are expected to be the key items for markets to watch," she added.
In corporate news, Melco International Development proposed a spin-off for its Macau resort Studio City, with plans to list the unit in the U.S., the company said in a filing to the Hong Kong Exchange Monday evening. Melco's stock was up 2.74 percent on the news, with most other Hong Kong-listed gaming plays turning negative in the second half of the day: Wynn Macau was up 1.85 percent, but Galaxy Entertainment erased earlier gains to trade 0.11 percent lower.
Also of note, the Commonwealth Bank of Australia may claw back bonuses from executives if required, the Australian Financial Review reported Monday, citing the bank's chairman Catherine Livingstone. CBA Chief Executive Ian Narev's bonus was scrapped last week as the lender deals with a money-laundering scandal.
CBA stock closed down 0.07 percent, as other Australian financials made gains: ANZ ended 1.29 percent higher after reporting a 5.3 percent rise in third-quarter cash profit while National Australia Bank advanced 0.59 percent by the end of the session.
Meanwhile, the Australian dollar sank after inching higher against the dollar after the Reserve Bank of Australia warned of the currency's strength and household debt levels in minutes of the central bank's latest policy meeting. The Aussie dollar traded at $0.7827 at 3:09 p.m. HK/SIN, after trading as high as $0.7876 earlier in the day.
Other market movers included Toshiba. The stock closed up 3.83 percent as Citi analysts upgraded the Japanese firm to "buy/high risk" from "neutral" in an Aug. 14 note. Toshiba shares were still a "risky investment," but the possibility of delisting has decreased, the note said.
Also concerning Toshiba, a California state court issued a preliminary injunction granting Western Digital access to data stored on Toshiba subsidiary Toshiba Memory Corporation's databases. Toshiba said it would comply with the order in a statement, adding that the move would not affect its memory business.
Still in Japan, Fujifilm shares closed 7.69 percent higher after the company reported record profit in the quarter ending in June on Monday, according to Reuters.
On the energy front, oil prices were mostly steady after falling more than 2.5 percent overnight. Brent crude edged up 0.02 percent to trade at $50.74 a barrel and U.S. West Texas Intermediate crude shed 0.08 percent to trade at $47.56. Oil fell on Monday as the dollar recovered and soft China demand data revived investor worries about oversupply in oil markets, according to Reuters.
— CNBC's Fred Imbert contributed to this report.