PARIS—One of the assailants in Friday’s deadly attacks in and near the French capital recently entered Europe as a Syrian migrant, people familiar with the matter said, exposing gaps in the continent’s security as it copes with the biggest refugee crisis in decades.

French investigators were able to match the remains of one of the suicide bombers who attacked France’s main stadium to a Syrian passport that was used to enter Europe’s free-travel area and apply for asylum, the people familiar with the matter said.

European...