The deal creates the world largest travel retailer, with a market share of 25% and projected annual sales of $9bn (€8.3bn; £6bn).
Dufry is paying €10.25 a share for the 50.1% stake, valuing the group at €3.6bn.
World Duty Free operates 495 stores in 98 airports, including London's Heathrow and Gatwick.
Based in Basel, Dufry has 1,650 stores in more than 60 countries with around 20,000 employees.
Last year it made a record turnover of $4.3bn. That compares to World Duty Free's turnover of $2.6bn, currently owned by Edizione, the holding company of the Benetton family.
Retail spending at airports is expected to rise in the years ahead, particularly in Asia where more than 350 new airports are set to be built.
Further details on the transaction are expected on Monday.
The Benetton family has sold its controlling stake in World Duty Free to Switzerland's Dufry.