The big win for Abe at Sunday's election for parliament's upper house meant he could now focus on implementing difficult economic reforms, the "Third Arrow" of his "Abernomics" prescription to end deflation. The other two arrows being super-easy monetary policy and spending.
But his resounding victory has also raised worries that he could shift focus to a nationalist agenda instead.
"The election results should be positive for the Nikkei and negative for the yen," JPMorgan analysts wrote in a report.
But they expected little market reaction given the outcome was already largely priced in, while debate around policies will not start until at least the end of the summer.
Tokyo's Nikkei was up 0.8 percent, having earlier climbed as much as 1.2 percent, while MSCI's broadest index of Asia-Pacific shares outside Japan .MIAPJ0000PUS advanced 0.4 percent.
"Abe's victory was widely expected, but it gives a positive lead to the market as it raises expectations that legislation will pass more easily and he can focus on revitalizing the economy," said Takuya Takahashi, a strategist at Daiwa Securities.
The currency market reacted with restraint to Abe's win, briefly selling the yen which plumbed a 1-1/2 week low on the dollar and a nine-week trough against the euro. It has since bounced back as the dollar softened broadly.
The greenback was last 0.8 percent lower on the day at 99.78 yen, reversing from an early high of 101.05, while the euro slipped 0.7 percent to 131.39 versus a high of 132.47.
Against the dollar, the euro drifted up 0.1 percent from late New York levels to $1.3166, while the Australian dollar advanced 0.5 percent to $0.9220.
The Aussie was further buoyed by news of more market-oriented reforms in the banking sector of China, Australia's single biggest export market.
China's central bank has removed controls on bank lending rates in a long-awaited move that could lower financial costs for companies, offering hopes that cheaper credit will help support the softening economy.
Commodities were mostly firmer thanks to the softer dollar. U.S. crude held near a 16-month peak of $109.32 a barrel, while copper gained 0.5 percent to $6,951 a tonne.
Gold reached a one-month high of $1,314.49 an ounce, continuing to recover from last month's eye-watering slide to a three-year low around $1,180.71.
There was little reaction to news that the Federal Reserve is "reviewing" a landmark 2003 decision that first allowed regulated banks to trade in physical commodity markets.
The one-sentence statement suggests the Fed is taking a much deeper, wide-ranging look at how banks operate in commodity markets than previously believed, amid intensifying scrutiny of everything from electricity trading to metals warehouses.
