The bullish sentiment provided the dollar with support, while the euro remains under pressure on expectations the European Central Bank will announce fresh measures to combat deflation fears.
Tokyo rose 0.24%, or 34.43 points, to finish at 14,670.95, Seoul gained 0.97%, or 19.43 points, to 2,017.06 and Sydney added 0.28%, or 15.5 points, to 5,527.2.
Shanghai climbed 0.77%, or 15.66 points, to 2,050.23 and Hong Kong added 0.59%, or 135.73 points, to 23,080.03.
Kuala Lumpur's main stock index added 4.09 points or 0.22% to 1,871.66, Singapore's Straits Times Index dropped 0.07%, or 2.22 points, to 3,271.84 and Jakarta closed up 0.44%, or 21.65 points, at 4,985.58.
Taipei put on 0.73%, or 66.42 points, to 9,121.71 and Manila rose 0.11%, or 7.62 points, to 6,787.88.
After the Memorial Day long weekend US shares rallied on Tuesday in response to more data showing the economy is getting back on track.
Independent business research group The Conference Board said its index of consumer confidence rose to 83.0 in May from 81.7 last month. It is also well up from 74.3 a year ago.
Home prices in the US rose 0.9% in March after a February stall caused by severe winter weather, the S&P/Case-Shiller index showed. And the Commerce Department said new orders for US durable goods rose 0.8% in April, a third consecutive pick-up and better than forecasts for a 1.3% fall.
The S&P 500 climbed 0.6% Tuesday, hitting another record after breaking the 1,900 level for the first time Friday.
The Dow added 0.42%, while the Nasdaq surged 1.22%.
Eyes are now on the release of revised first-quarter economic growth data on Friday.