STOCKHOLM—Sweden’s central bank cut its main interest rate to zero after similar attempts to boost inflation by central banks in the U.S., Japan and other parts of Europe, prompting the Swedish krona to sink to a four-year low against the euro and dollar.

The surprise move by the Riksbank on Tuesday underscores the challenge for smaller neighbors of the eurozone in grappling with weak inflation. The European Central Bank’s moves to bolster anemic prices in the currency bloc have caused currencies such as the krona to strengthen...