SEOUL—South Korea’s exports fell at their fastest pace since they were slammed by the fallout from the global financial crisis, the latest warning sign about international trade as concerns mount about the drag on growth from a slowing Chinese economy and withering oil prices.

Korean exports, the first shipments data released each month in Asia, slid 18.5% to $36.74 billion in January, the steepest fall since August 2009. The decline extended a run of monthly falls into a 13th month. Imports plunged 20.1% from a year earlier...