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The won followed a recovery in the yen, which had eased concerns over South Korea's weakening export competitiveness against Japan.
Malaysia's ringgit advanced as investors covered short positions. The ringgit also gained versus the neighbouring Singapore dollar.
The Chinese yuan rose against the dollar on Monday after the central bank set the midpoint at its strongest level since mid-March.
The dollar fell against a basket of six major currencies after data on Friday showing US employers added 214,000 new jobs to their payrolls in October, missing forecasts of 231,000.