Investors continue to see last week's US jobs figures as a sign the US Federal Reserve will not raise its rates until December at the earliest.
In Japan, the benchmark Nikkei 225 index closed up 1% to 18,186.10.
Investor sentiment was also buoyed in the region after a free trade deal was stuck between 12 Pacific Rim countries, the biggest in decades.
The US-led Trans-Pacific Partnership (TPP) cuts trade tariffs and sets common standards in member countries including Japan and the US.
Australia's S&P/ASX 200 benchmark closed 0.33% higher at 5,167.4.
Investors mostly shrugged off news that Australia's trade deficit rose unexpectedly in August to 3.09bn Australian dollars ($2.18bn; Â£1.44bn) against forecasts for a deficit of A$2.55bn.
The Reserve Bank of Australia (RBA) decided to keep its benchmark lending rate at an all-time low of 2% on Tuesday, but analysts said they would be looking for any hints of a December move when the minutes of the bank's meeting are released later this week.
However, market analyst Evan Lucas said: "We think the [bank's] statement will be even duller than the rates release [today] as the RBA will not want to give anything away.
"However, there is no escaping the fact that the market is building a case for a December cut, with the interbank market now pricing in a 60% chance that Australia will get a 25-basis-point cut [as a] Christmas bonus on 1 December."
In Hong Kong, the benchmark Hang Seng index closed down 0.1% to 21,831.62 after being up more than 1% in early trade.
The index was dragged down by shares of China Resources Enterprise, which fell 3.1%.
Stock markets in mainland China remain closed for the Golden Week holiday.
In South Korea, the Kospi index closed up 0.63% at 1,990.65.
Markets in Asia were largely in positive territory on Tuesday following a strong finish on Wall Street.