Appearing on a balcony over the square, he asked the faithful to pray for him. Cheers erupted as he gave a blessing.
The first Latin American and the first Jesuit to be pontiff, he will call himself Francis I.
An hour earlier, white smoke from the Sistine Chapel chimney announced the new Pope's election.
Pope Francis, 76, replaces Benedict XVI, who resigned last month at the age of 85, saying he was not strong enough to lead the Church.
French Cardinal Jean-Louis Tauran announced his election with the Latin words "Annuntio vobis gaudium magnum. Habemus Papam" ("I announce to you a great joy. We have a Pope").
"Habemus Papam Franciscum," was the first tweet by the papal account @pontifex since Benedict stood down last month.
The election was met with thunderous applause at the cathedral in Buenos Aires, Pope Francis' home city.
Throughout Latin America, people reacted with delight and surprise.
"It's a huge gift for all of Latin America. We waited 20 centuries. It was worth the wait," said Jose Antonio Cruz, a Franciscan friar in the Puerto Rican capital San Juan, quoted by the Associated Press.
Everyone from Canada down to Patagonia is going to feel blessed. This is an event."
Correspondents say Cardinal Bergoglio was a surprise choice and not among a small group of frontrunners before the election.
Many observers were also expecting a younger pope to be elected.
He is regarded as a conservative, but is known for his humility. He has spent almost his entire career in Argentina and often travels to work by bus.
The BBC's Marcia Carmo in Buenos Aires says Cardinal Bergoglio's sermons always had an impact in Argentina and he often stressed social inclusion and indirectly criticised governments that did not pay attention to those on the margins of society.
The name he has taken is reminiscent of St Francis of Assisi, the 13th Century Italian reformer and patron saint of animals, who lived in poverty.
The saint was said to have been summoned by God to repair a Church in ruins.
Cardinal Bergoglio is generally thought to have come second in the last conclave in 2005, which elected Benedict XVI as Pope.
The 115 cardinals involved in the 2013 election were in isolation since Tuesday afternoon, and held four inconclusive votes.
At least 77 of them, or two-thirds, would have had to vote for a single candidate for him to be elected Pope.
Before the conclave began, there appeared to be no clear choice to replace Benedict.
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