Foreign investors have purchased a net 1.85 trillion won ($1.8 billion) of the country's shares so far this year.
The Taiwan dollar gained as much as half a percent to 31.36 per dollar in local trading, its strongest since October 2000. Foreign investors were net buyers of local shares on Monday for the 13th straight session.
"There has been a real build-up of equity inflows over the last several sessions," said David Simmonds, emerging markets strategist at Royal Bank of Scotland in Singapore.
"The dollar is also weak against the yen. Sometimes you have this kind of opening of the floodgate situation. The direction is very clear."
Simmonds expected the won to breach 1,000 per dollar and strengthen to 980 in coming days. The Taiwan dollar could strengthen beyond 31 per dollar to about 30.70, he said.
"The Taiwan dollar has faster and further to go," he said. Analysts said that although the countries' central banks had intervened intermittently, their dollar-buying would only help smooth the rise in Asian currencies rather than reverse the trend.
Strong inflows of funds from overseas stock investors into Thailand has helped boost the baht by a third of a percent to about 38.45 per dollar.
The yen's 1 percent gain against the dollar since late Asian trading on Monday has also helped push up the Singapore dollar by 0.4 percent and both the Philippine peso and the Indonesian rupiah by a quarter of a percent.
"The market is going back to focus on the (dollar's) fundamentals, on the US current account deficit and on the relatively high-yielding currencies," said Simmonds.
Asian currencies were also buoyed by South Korea's announcement on Monday that it would diversify its $200 billion of foreign exchange reserves, Asia's fourth largest after Japan, China and Taiwan, away from government bonds into high-yielding paper and a greater variety of currencies.
South Korea's move follows similar announcements from Thailand, Taiwan and Indonesia. Asian governments are worried that the dollar's steep fall could cut the value of their reserves unless they diversify into currencies such as the euro.
Still, analysts at UBS noted that the Bank of Korea had given no indication it would actively sell the dollar. Instead, the central bank would put future reserves into higher-yielding securities and a wider range of currencies, they said.
UBS said it did not expect the South Korean won to gain much further because of its weak domestic economy, the likelihood of another cut in benchmark interest rates - which are already at a record low - and because of resistance to currency appreciation from the local authorities.
It expects the won to weaken to 1,040 per dollar in a month.