MSCI's broadest index of Asia-Pacific shares outside Japan added about 0.3%.
On Wall Street on Monday, US stocks turned in a solid performance, with the S&P 500 adding about 1%.
The United States and the European Union imposed sanctions, including asset freezes and travel bans, on a small group of officials from Russia and Ukraine after the weekend referendum.
"In immediate focus is Russian president Putin's speech later today. If he plays down an immediate annexation of Crimea by Russia, the dollar could gain further ground on unwinding of risk aversion," said Masafumi Yamamoto, chief strategist at Praevidentia Strategy in Tokyo.
"But an expression of desire for a swift annexation and retaliation against sanctions placed by the European Union and the United States will rekindle economic fears, driving US government bond yields lower and weighing on the dollar," he said.
Japan's Nikkei stock average rose 1.1%, recovering from a six-week closing low hit on Monday.
Short-covering was likely behind some of Tokyo's gains, as the ratio of short-selling to overall sales in the Tokyo Stock Exchange hit a record high of 36.17% the previous day.
"Some of those who sold yesterday are probably buying back today," said Hideyuki Ishiguro, senior strategist at Okasan Securities.
For the time being, risk appetite improved as the likelihood of immediate military conflict faded, and market participants turned their attention back to the US economic outlook and the conclusion of the Fed's two-day meeting on Wednesday.
The Fed is expected to continue to stick to reducing its monthly asset purchases by an additional $10 billion, and could also alter its forward guidance in its statement.
Fed policymakers could adopt less specific language to describe conditions under which it might tighten policy, instead of the bank's current threshold of a 6.5% unemployment rate for considering a rate rise. The rate now stands at 6.7%, though Fed officials are still signalling that rates need to stay low for some time to support the economy.
The dollar was slightly higher on the day at 101.78 yen, while the euro added about 0.1% to 141.80 yen.
The euro edged up about 0.1% to $1.3932, within sight of a 2-1/2-year high around $1.3967 touched on Thursday.
The single currency's resilience was despite data on Monday showing a dip in euro zone inflation, the latest indicator to back the view that the European Central Bank needs to take further monetary steps to support growth.
The improvement in risk sentiment took a toll on gold, which hit a six-month high on Monday before plunging more than 1%. It was last down about 0.4% at $1,360.21 per ounce, well shy of the previous session's peak of $1,391.76.
US crude edged down 0.1% to $97.97 a barrel, falling for the second day in a row, as expectations of growing petroleum stockpiles in the world's biggest oil user offset any fears that Ukraine tensions could worsen.