Traders also looked past a warning from the head of the Federal Reserve that a full recovery would likely not come until next year, and a vaccine would be needed to get things back to normal.
With infection and death rates falling in some of the worst-hit countries, governments are slowly allowing businesses to re-open and people to venture out again, with top-tier football returning in Germany -- albeit in empty stadiums.
California's governor said the state was 75% up and running, New York is also lifting the shutters in some regions, and Apple said almost 100 of its stores were now open.
"With the worst of the pandemic likely behind us, central bank-supported equity markets are unlikely to re-test their lows," said Seema Shah at Principal Global Investors.
But she added that "while reopening momentum may well carry risk assets a bit higher over the near term, the tepid economic recovery and deep uncertainty over the virus outlook argue against a pivot to more risk-on positioning".
Tokyo ended 0.5 percent higher, Hong Kong gained 0.6 percent and Shanghai closed 0.2 percent up.
Sydney jumped more than one percent, Singapore added 0.9 percent and Seoul was 0.5 percent higher, while there were also gains in Wellington, Bangkok and Jakarta.
But Manila dropped more than one percent and Mumbai shed almost three percent after the Indian government extended its lockdown until May 31.
In early trade, London rose two percent, Frankfurt jumped 2.2 percent and Paris climbed 1.8 percent.
"Good news for the market and the economy as a whole is that businesses worldwide are reopening, albeit in fits and starts," said AxiCorp's Stephen Innes.
"While restaurants are opening at minimal capacity and mall traffic remains depressed, traffic congestion is beginning to tick significantly higher, suggesting that people feel confident in leaving their homes.
"Indeed, this is huge as the global recovery will fall 100 percent on the back of consumer confidence."
The gains come despite a flurry of downbeat economic data, including Monday's news that Japan had fallen into its first recession since 2015.
While the January-March contraction was not as bad as expected, observers warned the current quarter would likely be much worse.
That followed a warning from Fed boss Jerome Powell that the US economy could "easily" collapse 20-30 percent this quarter, and unemployment could peak at 20 to 25 percent -- although he added "it should be a much shorter downturn than you would associate with the 1930s" during the Great Depression.
"Market reaction to these dire data releases was short lived, suggesting investors are now placing more value on forward looking indicators rather than real economic prints, which we all know have been negatively impacted by containment measures against COVID-19," said Rodrigo Catril at National Australia Bank.
Still, uncertainty over the economic outlook continues to play in the background, while there are also concerns about brewing tensions between China and the United States, which notched a little higher at the weekend when Washington ramped up sanctions on Huawei by cutting it off from global chipmakers.
Beijing responded by saying it would take "necessary measures" to protect the company and others.
OANDA analyst Jeffrey Halley said: "If Washington DC's intent to escalate trade issues has markets nervous, it is being trumped by the increasing pace of lockdown reopenings around the world, and the expected follow-on uptick in economic activity."
The reopenings continue to boost oil prices, with WTI -- which last month fell below zero -- breaking $30 for the first time since mid-March.
"There's been a very sharp reaction by US producers in cutting output and that's gone a long way in alleviating the stress on the system," said Daniel Hynes of ANZ, adding he did not see prices going back below $20 unless there was another mass wave of infections. - AFP
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