UEFA's ruling executive voted 7-6 for the French bid over Turkey after Italy was eliminated in a first-round ballot. France will be the first country to host the championship three times.
The result was announced by UEFA's French president Michel Platini, who captained his country to the European title in 1984 on home soil. Platini did not vote.
The close-fought victory came after Sarkozy and former France star Zinedine Zidane supported their country's case in the final presentations.
Sarkozy said the French government would help guarantee the projected $2.1 billion cost of building and renovating stadiums.
"The French state is here to support and guarantee that everything we have said, we will live up to," Sarkozy said.
The head of the French soccer federation, Jean-Pierre Escalettes, thanked UEFA for its confidence.
"This is a beautiful day for us," he said. "I can assure you that this trust will not be betrayed."
The 2016 championship will feature 24 teams playing 51 matches over one month.
The final will be played in the capital's Stade de France, which hosted the 1998 World Cup final won by the Zidane-inspired host nation.
France staged the inaugural four-team finals in 1960, and the eight-nation finals in '84.
The French bid focused on creating family-friendly stadiums as a legacy for future generations. The domestic game also has been troubled with outbreaks of fan hooliganism this season.
Sarkozy said he wanted the Euros to help French soccer be "free of the cancer of violence, a football with a human face and a football that is a festival and a party."
France's plans for 12 stadiums includes new venues in Bordeaux, Lille, Lyon and Nice; major renovations in Marseille and Strasbourg; as well as upgrades to existing arenas in Lens, Nancy, St. Etienne, Toulouse and Paris' Parc Des Princes.
Zidane spoke of being a young fan in the Marseille stadium watching Platini's extra time goal carry France past Portugal in a semifinal in 1984.
"That is the type of experience I want to live again," Zidane said. "Winning is beautiful but it's great to see all the people in the streets partying. This is what I would like to give my children."