South Korea's Kospi climbed 2.7 per cent and Taiwan's benchmark rose 1 per cent.
But Australia's index shed 0.4 per cent and Hong Kong's shares were 1.4 per cent lower by mid morning.
The Hang Seng had started Wednesday's session more than 2 per cent higher but large financial shares such as HSBC and Bank of China fell on worries about exposure to the debt of Lehman Brothers which filed for bankruptcy protection on Monday.
The Dow Jones had gained 1.3 per cent on Tuesday after falling on Monday to its lowest level since the September 11, 2001 attacks on the US.
Besides the AIG rescue, markets also appeared encouraged by the US Federal Reserve's decision to hold interest rates steady at 2 per cent - a move seen as a sign of confidence in a recovery.
The Asia upswing also comes as Japan's central bank said on Wednesday that it had injected two trillion yen ($19bn) more into money markets in an effort to stabilise financial markets.
The Bank of Japan had on Tuesday already released 2.5 trillion yen into the markets as other central banks collectively pumped hundreds of billions of dollars into money markets to ensure the supply of funds does not dry up.
AIG holds assets of more than $1 trillion
Has 74 million clients in 130 countries
Employs 116,000 people worldwide
Provides insurance, financial services and advice
Made $10bn profit just two years ago
Already lost $15bn this year
Ban Ki-moon, the UN secretary-general, expressed concern on Tuesday that the world's poorest could be worst hit by the financial crisis if rich donor nations lose their capacity to help achieve the UN millennium goals to fight poverty.
He urged a resumption of stalled world trade talks, which the UN sees as crucial to opening world markets to poorer developing countries.
Amy Auster, the head of international economics at ANZ Bank in Melbourne, Australia told Al Jazeera there was good news and bad news for the man in the street from the global financial turmoil.
The good news, she said, was that rising global inflation that had been a major worry just a few months ago, was now likely to come down, meaning food and fuel should become cheaper.
But economic growth was also slowing and that could mean a loss of many jobs.
Al Jazeera's John Terret explained that AIG was in trouble because it insured so many loans from private lenders to people who could not afford to meet monthly mortgage repayments.
Now all those people with insurance have come calling for their money and AIG - otherwise a sound business - cannot afford to pay them all.
The Federal Reserve said under the two-year loan deal to AIG, the government would receive a 79.9 per cent equity interest in the group and has the right to veto payment of dividends to shareholders.
"The board determined, that, in current circumstances, a disorderly failure of AIG could add to already significant levels of financial market fragility and lead to substantially higher borrowing costs, reduced household wealth and materially weaker economic performance," the US central bank said in a statement.
It said that the loan was "expected to be repaid from the proceeds of the sale of the firm's assets" and try to assure Americans that "the secured loan has terms and conditions designed to protect the interests of the US government and taxpayers".
Henry Paulson, the US treasury secretary, had just a day ago called for "market discipline" to let failing institutions fail in the wake of the collapse of US investment bank Lehman Brothers, indicating that he was wary of more taxpayer-funded bail-outs of major financial institutions.
But he said he backed the AIG deal.
"These are challenging times for our financial markets," he said in a statement. "I support the steps taken by the Federal Reserve tonight to assist AIG in continuing to meet its obligations, mitigate broader disruptions and at the same time protect the taxpayers."
Federal Reserve officials told reporters on Tuesday that the AIG rescue was necessary, unlike that of Lehman Brothers, because of the insurer's extensive ties to other firms and retail products.
Markets were more prepared for Lehman's failure and the Securities and Exchange Commission had procedures in place to deal with a securities firm's failure, the officials said, but AIG was deemed to be a very complex firm with extensive operations in many parts of the financial markets including retail financial products, such as insurance and guaranteed annuities, they added.
The fallout from AIG's collapse could affect not just investors but the average person who buys such products as car or life insurance.
Far more than other insurers, AIG has been a big player in a complex parallel market called credit default swaps, financial instruments in which Wall Street companies take out a form of market insurance against the risks of bond default.
These products, often linked to the US real estate market, are at the heart of the current banking crisis and have led to massive write-downs of assets around the world.