By Agence France-Presse
The US president's comments provided a much needed boost to investors after a month of volatility sparked by his shock decision to hit China with fresh tariffs, ending months of apparently positive negotiations.
Adding to the upbeat mood were comments from the European Central Bank head Mario Draghi hinting at a cut in interest rates to support the stuttering eurozone economy.
Also, the Federal Reserve is due to end its latest policy meeting later Wednesday, with dealers hoping for some idea about its own plans for rates.
After all three of Wall Street's main indexes posted gains of one percent or more, Asia picked up the baton, with Hong Kong piling on more than two percent, Tokyo 1.7 percent, Shanghai and Singapore 1.4 percent, and Sydney one percent.
Taipei, Wellington and Seoul were also more than one percent higher.
The rally was sparked after Trump tweeted: "Had a very good telephone conversation with President Xi of China. We will be having an extended meeting next week at the G-20 in Japan. Our respective teams will begin talks prior to our meeting."
Later he told reporters "the meeting might very well go well", adding the China wanted to make a deal.
"China and the US will both gain by cooperating and lose by fighting," Xi told Trump, according to a readout by Chinese state broadcaster CCTV.
Trump's tweet followed weeks of speculation about whether the heads of the two most powerful economies would actually meet on the sidelines of the G20 in Osaka. Trump had warned that if Xi did not turn up he would hike tariffs on virtually all China's exports to the US.
However, analysts pointed out that it was in both leaders' interests to bring an end to the long-running dispute.
"There is strong incentive for both presidents to re-engage," said Tai Hui, chief market strategist for Asia-Pacific at JP Morgan Asset Management.
"Trump is kick-starting his (re-election) campaign and he will need strong economic performance over the next 18 months. President Xi will also need trade tension to cool down to support China's domestic economy, while pursuing financial market liberalisation."
The optimism underpinned a rally in riskier assets, with high-yielding currencies benefiting. South Korea's won jumped 0.7 percent, the South African rand added 0.8 percent and Indonesia's rupiah gained 0.5 percent.
The Chinese yuan, which has struggled in recent weeks, climbed 0.4 percent.
The euro, however, extended Tuesday's losses after Draghi's remarks that weak growth and soft inflation could lead to further rate cuts to historic lows.
He also batted off an accusation from Trump of currency manipulation, saying the ECB's mandate "is price stability".
Oil prices also pushed ahead after surging on hopes that a resolution to the trade war would boost demand for the commodity. Brent climbed almost two percent and WTI almost four percent Tuesday.
Key figures around 0230 GMT
Tokyo - Nikkei 225: UP 1.7 percent at 21,321.17 (break)
Hong Kong - Hang Seng: UP 2.3 percent at 28,129.71
Shanghai - Composite: UP 1.3 percent at 2,928.69
Euro/dollar: DOWN at $1.1197 from $1.1202 at 2040 GMT
Pound/dollar: UP at $1.2561 from $1.2558
Dollar/yen: UP at 108.59 yen from 108.43 yen
Oil - West Texas Intermediate: UP 23 cents at $54.34 per barrel (new contract)
Brent North Sea: UP 19 cents at $62.33 per barrel
New York - Dow: UP 1.4 percent at 26,465.54 (close)
London - FTSE 100: UP 1.2 percent at 7,443.04 (close)