A member of the Agnelli family, Elkann was born in New York City, the son of Countess Margherita Agnelli de Pahlen and the American-born Italian writer Alain Elkann. His father is Jewish and his mother is Catholic. He is the brother of John Elkann, the current Chairman of Fiat and the Chairman and CEO of Exor, an investment company controlled by the Agnelli family, which owns Juventus F.C., Cushman & Wakefield. Elkann is a grandson of Fiat chairman Gianni Agnelli and Marella Agnelli. He is nephew of Don Carlo Caracciolo, founder of Gruppo Editoriale L’Espresso - owner of the newspaper la Repubblica, the Italian daily general-interest newspaper. Elkann is also a stepson of Russian nobleman Count Sergei de Pahlen.
In addition to his brother John, Elkann has a sister, Ginevra Elkann, and five half-siblings by his mother's second marriage: Maria, Pierre, Sophia, Anna, and Tatiana. He is also a great-grandnephew of Ettore Ovazza, of the Ovazza banking family. Elkann descends from the Ovazza banking family, via his grandmother, Carla Ovazza.
In 2001, Elkann worked as a personal assistant to Henry Kissinger, subsequently working as marketing associate for Ferrari and Maserati. In 2003, he became Fiat's Director of Marketing. He coordinated marketing for the Fiat Punto and Fiat 500 and received the America Award of the Italy-USA Foundation in 2011.
Elkann subsequently left Fiat, creating Italia Independent, a lifestyle and clothing brand featuring Italian products. In 2007 he founded Independent Ideas, a communication and advertising agency.
Since 2011 he has consultted with Ferrari, running Ferrari Tailor Made, which enables customers to customize their car.
In June 2013 Lapo listed his company Italia Independent on the Milan Stock Exchange, Alternative Listing; the funds will be used to expand sales of his eyewear stock in Europe and Switzerland.
Vanity Fair magazine listed Elkann in its Best Dressed Hall of Fame of 2009 and was named to the magazine's on the 2008 International Best Dressed List. He received the America Award of the Italy-USA Foundation in 2011.