Pope Francis, 76, is also the first member of the Jesuit order to be Pope, and the first Latin American. He made his first appearance on the central balcony of St Peter's Basilica in front of thousands of people, while millions around the world watched on TV.
His appearance came about an hour after white smoke rose from a chimney on the roof of the Sistine Chapel to signal a Pope had been chosen to lead the world's 1.2 billion Catholics.
Pope Francis was elected on the fifth ballot, on the second day of one of the fastest recent conclaves, surprising many, including the leader of Scotland's Catholics.
A trumpet fanfare heralded the new Pope's entrance for his first official address as the thousands of followers gathering below chanted: "Viva il Papa!"
Dressed in his fresh white robes, the bespectacled Argentinian prelate initially looked pensive and even intimidated as he surveyed the ocean of jubilant visitors in rain-soaked St Peter's Square.
"Pray for me," said the new pontiff, urging the crowd whose chants turned to "Francesco! Francesco".
In a light-hearted moment he said his fellow cardinals had gone to the "ends of the Earth" to find him and added: "Here I am."
He prayed for his predecessor, the recently retired Benedict XVI, before inviting the faithful to "pray for the entire world". He added: "I hope that this path for the Church will be one fruitful for evangelisation."
The temporary leader of Scotland's Catholics, Philip Tartaglia, Archbishop of Glasgow, was "surprised the conclave was so quick". He said: "Given the pre-conclave situation, and the absence of a single dominant figure, I can only see this quick result as God's work.
"As soon as I heard I went into my chapel and offered a grateful and joyful prayer of thanksgiving - Deo Gratias!
"I think we can take from his first appearance that he is a humble, spiritual and calm man, a reconciler and healer, with a strong background on social justice."
The choice of Bergoglio was announced by French Cardinal Jean-Louis Tauran with the Latin words "Annuntio vobis gaudium magnum. Habemus Papam" which means "I announce to you a great joy. We have a Pope."
The election was met with loud applause at the cathedral in Buenos Aires, Pope Francis's home city.
Pope Francis replaces Benedict XVI, who resigned last month at the age of 85, saying he was not strong enough to lead the Church.
It is believed taking the papal name of Francis is an early indication of a reign he hopes will be marked by inspirational preaching and evangelisation.
The new pontiff telephoned Benedict and is planning to meet him, a Vatican spokesman said.
"Habemus Papam Franciscum" was the first tweet by the papal account @pontifex since Benedict stood down last month.
Pope Francis, who initially trained as a chemist before becoming a priest, becomes the 266th pontiff at a time of grave crisis for the Church, which is under fire over child sex abuse. Although a conservative, he is seen as a reformer.
He has spoken out strongly against gay marriage, calling it "an attempt to destroy God's plan" in 2010.
It is thought the election of the son of an Italian immigrant to Argentina helped reconcile two conflicting trends in the papal conclave: the push to return to the tradition of Italian popes, and the longing for a pontiff from the developing world.
His election also overturned one of the main assumptions before the election, that the new Pope would be relatively young.
Bergoglio is understood to have received the second highest number of votes when Pope Benedict was elected. According to some accounts, however, he was not chosen eight years ago because he asked his fellow-cardinals not to continue voting for him.
US President Barack Obama sent "warm wishes" on behalf of the American people to the newly elected pontiff, hailing him as "the first Pope from the Americas".
Prime Minister David Cameron tweeted: "A momentous day for the 1.2bn Catholics around the world as His Holiness Pope Francis I is appointed the 266th Bishop of Rome."
Commenting on the election of the new Pope in Rome, First Minister Alex Salmond said: "My warmest congratulations go to Pope Francis on his election, which will be welcomed by Scotland's Catholic community and by others around the world."
But the cardinals' choice risked running into immediate controversy over the new Pope's role in Argentina's troubled history and an apparent failure to openly challenge Argentina's dictatorship of 1976 to 1983. In his book, El Silencio, a prominent Argentinian journalist alleged that the then Jorge Bergoglio connived with the Argentine navy to hide political prisoners of the military dictatorship from a visiting delegation of the Inter-American Human Rights Commission.
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