Japan's Nikkei 225 index advanced more than 1.8 percent, while Hong Kong's Hang Seng index rose about 0.7 percent in mid-afternoon trading. Australia's S&P/ASX 200 index added 0.65 percent and South Korea's Korea Composite Stock Price Index, or KOSPI, climbed about 1.4 percent.
European and U.S. stock indexes were expected to rise Monday after finance ministers from the 17 countries that use the euro currency worked out the bailout deal, which beat the European Central Bank's Monday deadline with only hours to spare, keeping the island in the eurozone but likely hurting Cyprus' economy.
"We have a deal," Cypriot President Nicos Anastasiades said after the early-morning deal was announced.
He called the deal "in the interests of the Cypriot people and the European Union."
Cypriot Parliament President Yiannakis Omirou, a member of the main opposition party, called the deal a "positive" development.
The deal, whose details were still being released, shuts Cyprus Popular Bank PCL, or Laiki Bank, the island nation's second-largest but most-troubled bank.
Bank depositors with accounts worth less than 100,000 euros ($130,000) would not be touched but deposits of 100,000 euros or more would be hit with a loss.
The level of losses for large depositors would be announced later Monday, when European Union and International Monetary Fund experts run their calculations, The Wall Street Journal reported.
Some of Laiki's assets and liabilities will be transferred to the Bank of Cyprus PCL, the country's largest bank, which will survive as a much smaller entity, officials said.
The move will create losses to be borne by large Bank of Cyprus depositors, officials said. The amount of the losses would also be announced later, but could be as high as 40 percent, The Financial Times reported.
The losses would be in return for equity shares in the reformulated bank, British newspaper The Guardian reported.
Cyprus was promised 10 billion euros ($13 billion) in aid from the eurozone if it could come up with 5.8 billion euros ($7.6 billion) on its own.
The bank restructuring doesn't need approval by the Cypriot Parliament, German Finance Minister Wolfgang Schauble said, because the Parliament passed a bill Friday allowing a fundamental restructuring of the banking sector.
But the deal "will lead to fundamental changes" in the Cypriot banking sector, he said.
Officials said they believe the country will need strict controls on money transfers in and out of the economy, for weeks if not months, the Journal said.
The Cypriot economy could shrink 10 percent or more in the years ahead, officials and economists told the newspaper.
Russians -- estimated to hold more than $26 billion of the $89 billion deposited in Cypriot banks -- were expected to lose billions under the deal's terms.
Cyprus had styled itself as a tax haven to attract international deposits.
Schauble said the troika of official lenders -- the eurozone, International Monetary Fund and the European Central Bank -- would be "in contact with the Russian government."