The 90,000-seat stadium, which staged the Champions League finals in 2011 and 2013, beat off a rival bid from Munich for the final three matches of the tournament.
Nineteen cities were bidding to stage matches during the tournament after UEFA's decision to spread the championship around 13 cities across the continent, as against previous versions staged in one or two countries.
Twelve cities were awarded standard packages, comprising three group games and a knockout clash.
Quarter-final matches will be played in Baku, Munich, Rome and St. Petersburg with last 16 ties in Copenhagen, Amsterdam, Bucharest, Dublin, Bilbao, Budapest, Brussels and Glasgow.
England has only hosted one major tournament since the 1966 World Cup, the Euro 96 tournament, and suffered the disappointment of failed bids for the 2006 World Cup, losing out to Germany, and the 2018 World Cup to be hosted by Russia.
"This bidding process was open to more than 50 UEFA countries so for Wembley to be ultimately recognised in this way is testament to a lot of hard work behind the scenes," England's FA chairman Greg Dyke said.
"It will be a great honour to be part of what will surely be a superb celebration of 60 years of the UEFA European Championship."
Six cities, Cardiff, Stockholm, Skopje, Jerusalem, Sofia and Minsk, were left disappointed.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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