He is the first ever from the Americas and the first from outside Europe in more than a millennium.
The 76-year-old, from Buenos Aires, has chosen to be known as Pope Francis.
A stunned-looking Bergoglio shyly waved to the crowd of tens of thousands of people who gathered in St Peter's Square, marvelling that the cardinals had had to look to "the end of the earth" to find a bishop of Rome.
He asked for prayers for himself, and for retired Pope Benedict XVI, whose stunning resignation paved the way for the tumultuous conclave that brought the first Jesuit to the papacy.
The cardinal electors overcame deep divisions to select the 266th pontiff in a remarkably fast conclave.
Bergoglio had reportedly finished second in the 2005 conclave that produced Benedict - who last month became the first pope to resign in 600 years.
After announcing "Habemus Papum" - "We have a pope!" - a cardinal standing on the balcony of St Peter's Basilica revealed the identity of the new pontiff, using his Latin name.
The 76-year-old archbishop of Buenos Aires has spent nearly his entire career at home in Argentina, overseeing churches and priests.
Tens of thousands of people who braved cold rain to watch the smokestack atop the Sistine Chapel jumped in joy when white smoke poured out, many shouting "Habemus Papam!" or "We have a pope!" - as the bells of St Peter's Basilica and churches across Rome pealed.
Chants of "Long live the pope!" arose from the throngs of faithful, many with tears in their eyes. Crowds went wild as the Vatican and Italian military bands marched through the square and up the steps of the basilica, followed by Swiss Guards in silver helmets and full regalia.
They played the introduction to the Vatican and Italian anthems and the 50.000 strong crowd joined in, waving flags from countries around the world.
Elected on the fifth ballot, Francis was chosen in one of the fastest conclaves in years, remarkable given there was no clear front-runner going into the vote and that the church had been in turmoil following the upheaval unleashed by Pope Benedict XVI's surprise resignation.
A winner must receive 77 votes, or two-thirds of the 115, to be named pope.
For comparison's sake, Benedict was elected on the fourth ballot in 2005 - but he was the clear front-runner going into the vote. Pope John Paul II was elected on the eighth ballot in 1978 to become the first non-Italian pope in 455 years.
First Minister Alex Salmond was among the first to offer congratulations. He 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.
"I wish him well in his time in the Papacy, and his first message as Pope - urging greater bonds of understanding and co-operation among peoples and nations - is one that should resonate around the globe."
Jorge Mario Bergoglio has spent nearly his entire career at home in Argentina, overseeing churches.
His election might help to reconcile two conflicting trends in the papal election: the push to return to the tradition of Italian popes, and the longing for a pontiff from the developing world.
Cardinal Bergoglio, 76, reportedly got the second-most votes after Joseph Ratzinger in the 2005 papal election, and he has long specialised in the kind of pastoral work that some say is an essential skill for the next Pope.
Bergoglio is known for modernising an Argentine church that had been among the most conservative in Latin America.
He stands out for his austerity. As Argentina's top church official, he's never lived in the ornate church mansion in Buenos Aires, preferring a simple bed in a downtown room heated by a small stove on frigid weekends. For years, he used public transport around the city, and cooked his own meals.