Shares in mainland China saw solid gains on the day following the positive developments. The Shanghai composite surged 5.6 percent to 2,961.28 while the Shenzhen component added 5.587 percent to 9,134.58. The Shenzhen composite jumped 5.417 percent to 1,557.27.
Those moves took the Shanghai composite into bull market territory, or up at least 20 percent from intraday lows seen in early January. The Shanghai composite fell into a bear market, or down 20 percent from a recent high, in June 2018.
Over in Hong Kong, the Hang Seng index rose 0.52 percent in its final hour of trading. Shares of China Construction Bank gained more than 2.1 percent. Hong Kong-listed shares of Chinese network equipment firm ZTE advanced more than 2.3 percent, after leaping 13.9 percent earlier, according to Reuters.
Chinese markets in general have posted strong gains so far in 2019, the Shanghai composite, for example, is up more than 18 percent year-to-date and even the smaller CSI300 has jumped more than 20 percent year to date. That surge since the start of 2019 led investment bank UBS to caution investors that the market has gone "a long way ahead of fundamentals."
In Japan, the Nikkei 225 advanced 0.48 percent to close at 21,528.23 and the Topix rose 0.71 percent to finish its trading day at 1,620.87. Shares of index heavyweight Fast Retailing, the company behind the Uniqlo chain of apparel stores, gained 0.67 percent.
Australia's also added 0.31 percent to close at 6,186.30, while South Korea's Kospi finished its trading day fractionally higher at 2,232.56.
The broad MSCI Asia-ex Japan index rose 0.63 percent to 527.85 as of 3:16 p.m. HK/SIN.
"As both (the U.S. and Chinese) Presidents said significant progress has been made, the chance for the US and China to reach a deal is getting higher," OCBC Treasury Research said in a morning note. "However, the key to medium term stability hinges on two areas including agreement on China's structural reforms and the enforcement of trade deals."
U.S. President Donald Trump announced in a series of posts on Twitter Sunday evening that America is planning to delay a set of additional tariffs on Chinese goods that were due to kick in on March 1.
In his posts, Trump cited "substantial progress" in bilateral talks between the world's two largest economies, including intellectual property protection and technology transfer issues. As a result, the president said he would suspend the new levies, however he did not state a new deadline.
"President Trump announcing a delay in raising tariffs on Chinese imports has been in the works for some time. The high frequency engagement between Beijing and Washington at a senior level implies that both sides are looking for some form of settlement," Tai Hui, Asia Pacific chief market strategist at J.P. Morgan Asset Management, said in a note.
"Moreover, with growing questions over the growth outlook in the U.S., further tariff escalation would add more uncertainty to this concern. Still, there are plenty of long term structural issues for both sides to lock horns over, such as market access and (intellectual property rights) protection," he said.
The U.S. president's comments came following extended trade negotiations between officials from Washington and Beijing. Talks had earlier spilled over into the weekend after both sides reported progress in narrowing their differences.
Last week, sources familiar with the situation told CNBC that the United States and China are discussing a late March meeting between Trump and Chinese President Xi Jinping in Florida.
The U.S. dollar index, which tracks the greenback against a basket of its peers, was at 96.412 after touching highs above 96.9 last week.
The Japanese yen, often viewed as a safe-haven currency, traded at 110.66 against the dollar after touching lows above 110.8 last week. The Australian dollar changed hands at $0.7147 after seeing turbulence last week as it swung between highs above $0.720 and lows below $0.710.
The offshore Chinese yuan also saw gains, on the back of Trump's announcement, last trading at 6.6936 against the dollar. Its changed hands at 6.6923 against the greenback.
Meanwhile, oil prices shed their earlier gains to see losses in the afternoon of Asian trade as the international benchmark Brent crude futures contract declined 0.18 percent to $67.00 per barrel. The U.S. crude futures contract also fell fractionally to $57.21 per barrel.
— Reuters, along with CNBC's Javier E. David and Evelyn Cheng, contributed to this report.