Equities in Japan, Australia, Singapore, Taiwan and Jakarta surged, while markets in Hong Kong, China and South Korea were closed for a public holiday.
London, Paris and Frankfurt also opened higher on Monday.
The euro continued to gain against the dollar, with investors in Europe weighing the prospect of the European Central Bank (ECB) following the US Federal Reserve's lead and raising key rates.
On Sunday, German central bank president Joachim Nagel signalled the ECB would probably continue raising rates to curb runaway inflation.
Nagel predicted inflation in Europe might peak at more than 10 percent in December.
The ECB raised the key rate by a historic 75 basis points last week, and markets expect a similar-sized hike at an October meeting.
This week, investors worldwide will be closely watching US inflation data for August, due to be released on Tuesday, with the consumer price index (CPI) expected to ease slightly to eight percent -- still well above the Fed's two-percent target.
Traders expect the Fed to impose another large rate hike next week, after two 75-basis-point increases already.
"A downside surprise in US CPI is likely more of a concern and that could see the dollar weakening further," Charu Chanana, a strategist at Saxo Capital Markets, told Bloomberg Television.
Clifford Bennett, chief economist at ACY Securities, said he expected stocks to "continue to drift higher" ahead of Tuesday's US CPI data.
"(US CPI) may well see further improvement as petrol prices have continued to pull back," he said.
"Other components are still likely to be pointing higher, but fuel prices could well dominate this CPI number."
Oil began the week with a slump, as investors weigh the possibility of global demand weakening as growth slows and China's harsh zero-Covid policy continues to sap economic activity.
On Monday, new data showed British GDP expanded by 0.2 percent in July, according to the Office for National Statistics.
Concerns remain, however, about the overall health of the UK economy.
"July's GDP remains below the level seen in May, pointing to an overall contraction over the first two months of summer," said Yael Selfin, chief economist at KPMG UK.
"Soft landing" hopes
US Treasury Secretary Janet Yellen on Sunday said she was hopeful the US economy could avoid a recession, but that the Fed would need to skilfully manage interest rates and also rely on "some good luck to achieve what we sometimes call a soft landing".
"My hope is we will achieve a soft landing, but Americans know it's essential to bring inflation down and, over the longer run, we can't have a strong labour market without inflation under control," she told CNN.
Yellen said that while the US economy's growth rate was slowing, the labour market remained "exceptionally strong", with almost two openings for every jobseeker.
In addition to the US CPI figures on Tuesday, traders will be closely watching UK CPI on Wednesday, and European CPI and China home sales, retail sales and industrial production data on Friday.
In Tokyo, stocks closed higher on Monday with gains by tech shares and a weaker yen boosting the market.
The dollar fetched 143.18 yen in Asian trade, against 142.56 yen on Friday in New York.
"A cheaper yen is positive for corporate performances, despite recent media reports" that highlight the negative aspects of the weak yen, said chief strategist Masayuki Kubota of Rakuten Securities.
On Friday, Bank of Japan chief Haruhiko Kuroda met Prime Minister Fumio Kishida, saying the rapid weakening of the currency was "undesirable", an indication of possible upcoming action to arrest the fall.
Key figures at around 0700 GMT
Tokyo - Nikkei 225: UP 1.2 percent at 28,542.11 (close)
Hong Kong - Hang Seng Index: closed for public holiday
Shanghai - Composite: closed for public holiday
New York - Dow: UP 1.2 percent at 32,151.71 (close)
New York - S&P 500: UP 1.5 percent at 4,067.36 (close)
New York - Nasdaq: UP 2.1 percent at 12,112.31 (close)
London - FTSE 100: UP 0.4 percent at 7,381.19
Frankfurt - DAX: UP 0.6 percent at 13,167.26
Paris - CAC 40: UP 0.3 percent at 6,228.02
EURO STOXX 50: UP 0.4 percent at 3,583.28
Euro/dollar: UP at $1.0147 from $1.0046
Pound/dollar: UP at $1.1661 from $1.1587
Euro/pound: UP at 87.00 pence from 86.84 pence
Dollar/yen: UP at 143.18 yen from 142.56 yen
Brent North Sea crude: DOWN 0.52 percent at $92.34 per barrel
West Texas Intermediate: DOWN 0.73 percent at $86.16 per barrel