The benchmark index closed up 7.71% at 18,770.51 points. A day earlier, the index saw all the gains it had made this year wiped out.
Remarks by newly re-elected prime minister Abe suggesting company tax cuts were on the way helped the mood.
Also positive were Tuesday's rebound for US shares and an improving Chinese share market.
Investor sentiment was up across the rest of Asia.
Tuesday's weak economic data from China has also raised hopes of more stimulus for that economy and its markets.
Hong Kong's benchmark Hang Seng index finished up 4.1% at 22,131.31 - marking its biggest one-day percentage gain in almost four years.
The Chinese government said on Wednesday that it would strengthen fiscal policy, boost infrastructure spending and speed up reform of its tax system to support the economy.
On the mainland, the Shanghai Composite closed up 2.3% at 3,243.09 - moving into positive territory for the year.
In Australia, the S&P/ASX 200 closed up 2.07% at 5,221.10, taking its lead from US markets.
Analysts said resource and commodity shares, together with some of the big bank stocks, had buoyed the Australian index.
Numbers out on Wednesday showed consumer confidence slid in September which led to revived hopes of another rate cut by the Reserve Bank of Australia.
South Korea's Kospi benchmark index also closed up 2.96% at 1,934.20 points. Official data released on Wednesday showed the country's latest unemployment figures for August sitting at their lowest since January this year.
Chris Weston from IG Markets said it would be a good news day for markets and that there was a "sea of green on screen in risk associated assets".
"This pick-up in sentiment once again started from a nice move higher in S&P futures during Asia, helped on by some really bullish flow in the Chinese markets," he said in a note.
"Specifically, the H-shares (Chinese dual-listed companies trading in Hong Kong) had its best day [on Tuesday] in months," he added.
"In the Chinese mainland, there has been some focus on headlines on a 50% reduction in personal income tax dividends for larger shareholders, with the idea being to move the market away from short-term and focus on the longer-term."
Japan's Nikkei surged almost 8% on Wednesday in its biggest one-day jump since late 2008.