Like Alexander the Great whose sword cut the Gordian knot, Margrethe Vestager has sliced through what until now has appeared to be an intractable problem: multinational companies earning billions of dollars in Europe and paying tiny sums in taxes to European governments.

The European Union’s competition chief ordered Ireland to recoup $14.5 billion in taxes from Apple Inc., a sum that is so large that it will light a fire under multinationals’ tax planning. The effects will reverberate far more quickly than the slow-moving...