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Tokyo's Nikkei advanced 0.7% on its last trading day of the year. The market is closed from Tuesday to Friday. Australian stocks rose 0.6% to bring their gains for the year to 15%.
Much of Asia, however, continued to underperform, in part due to investors shifting funds from emerging markets and into Europe and the United States.
Japan's competitors have also been complaining about the weak yen giving it a trade advantage. South Korea's deputy finance minister warned the yen was falling too fast, and the head of China's National Development and Reform Commission said the impact on neighbours needed to be monitored.
That could have been one reason MSCI's broadest index of Asia-Pacific shares outside Japan was only up 0.1%, and looked set to dip 0.2% for the year.
In stark contrast, Japan's Nikkei has risen 56.7% in 2013, its best annual performance since 1972, urged on by aggressive monetary and fiscal stimulus.
There were more promising signs for the economy when the Asahi newspaper reported Japan's most influential business lobby has agreed to encourage its members to raise workers' base pay for the first time in six years.
Many economists say an increase in base pay is essential to Prime Minister Shinzo Abe's pledge to end 15 years of deflation and to help the Bank of Japan meet its 2% inflation target.
Aiding the economy has been the fall in the yen this year, which has left it at five-year trough against the dollar and euro.
The dollar was up at 105.36 yen on Monday after reaching a fresh peak at 105.415. The yen has posted ninth consecutive weeks of falls against the dollar, the longest such run since 1974.
The euro was also firm at 144.705 yen, having been as far as 145.67 yen on Friday.
Thin year-end conditions made for some wild moves, with the euro vaulting as high as $1.3892 on Friday before falling back. On Monday, the single currency was somewhat calmer at $1.3740 with offers crowded in the $1.3810/35 area.
The single currency could find further support from comments by European Central Bank President Mario Draghi that he saw no urgent need to cut interest rates again and no signs of deflation.
Financial bookmakers expected Britain's FTSE 100, Germany's DAX and France's CAC 40 to open steady to modestly softer.
Less positive was news that Italy's third-biggest bank, Monte dei Paschi di Siena was forced to delay a vital 3 billion euro share sale because of shareholder opposition, plunging its turnaround plan into uncertainty.
The world's oldest bank needs to tap investors for cash to pay back state aid and avert nationalisation.
Underpinning both the dollar and euro have been widening yield premiums over Japanese debt.
Yields on the US benchmark 10-year Treasury note have climbed to their highest in more than two years at 3.02%. The comparable Japanese yield is just 0.735%.
Analysts at RBS note that yields on the 30-year Treasury bond were approaching a hugely important level at 4.05%, which marks the top of a bull channel going back two decades. A breach there would be viewed as very bearish for bonds.
In commodity markets, London copper was up at its highest level in four months, with signs of economic revival in Asia and the United States burnishing the demand outlook for industrial metals.
Gold edged down to $1,206 per ounce and was on track for its biggest annual loss in three decades at nearly 30%.
Brent crude oil was 46 cents firmer at $112.64 a barrel, while US light sweet crude inched up 5 cents to $100.37 a barrel.
