President Michael D Higgins has conveyed his sincere best wishes to the new head of the Catholic Church.
Jorge Bergoglio, an Argentine cardinal of Italian descent, was elected Pope on the conclave's fifth ballot.
The Jesuit will take the name Pope Francis In his message, President Higgins conveyed the best wishes of the people of Ireland to the 76-year-old as he assumed the office of Holy Father.
"Despite the pressures and distractions of a more complex world, people all over the globe still yearn for a sense of spiritual meaning to their lives," said Mr Higgins.
"Your leadership of the church will therefore be of immense importance in providing ethical guidance and moral encouragement towards a world that is more just, more compassionate and more focused on a shared sense of community.
"While your election to this challenging ministry of service brings great personal responsibilities, I sincerely hope it will also offer opportunities for peace and joy as you carry out your duties in solidarity with the wider church community."
Mr Higgins said the election will bring great joy to the people of Argentina and of South America, as well as to the Jesuit community worldwide.
"I look forward to meeting you in person at an appropriate opportunity in the future," the president added.
Cardinal Bergoglio, who was regarded as a surprise choice, is the first Jesuit, as well as the first from the Americas and the first from outside Europe in more than a millennium.
His dramatic election was signalled when white smoke billowed from the conclave of cardinals meeting in the Sistine Chapel in Rome.
Cardinal Bergoglio had reportedly finished second in the 2005 conclave that voted Pope Benedict XVI, who last month became the first pope to resign in 600 years.
Archbishop of Dublin Diarmuid Martin said the new Pope is a man who lives simply, cares for the sick and poor, and kept his distance from the military and Government.
"He's a real pastor," he said.
"The name Francis is a clear indication that he has a very particular view of what the church and simplicity and poverty is about."
The archbishop said Cardinal Bergoglio spoke to him once at a synod about the difficulties and challenges the church in Ireland faced in the wake of the clerical child abuse scandals.
"He said very explicitly 'your challenges in Dublin, my challenges in a large city like Buenos Aires are very similar' so his interest is being a pastor," Archbishop Martin said.
"He was extraordinarily supportive."
Taoiseach Enda Kenny said the scenes of celebration in St Peter's Square are a reflection of the great hopes and best wishes of people of the Catholic faith all around the world have for the new pope.
"On behalf of the Government and the people of Ireland, I offer warm congratulations to Cardinal Jorge Mario Bergoglio on his election as Pope Francis," he said.
"He has the best wishes of all Irish people, of all traditions, as he undertakes the immense responsibility of his pontificate.
"We pray that he will have the strength, the good health and the spiritual guidance needed to lead the Catholic Church in the many challenges it faces."
Mr Kenny said the decision of the conclave in choosing the first pope from the Americas is of particular significance as a symbolic expression of the global nature of the Catholic Church and of the 21st century world.
"The people of Ireland also send our congratulations to the people of Argentina, a country with which Ireland shares many historic bonds, on this occasion of great honour and pride for their nation," he added.
US President Barack Obama also offered warm wishes to new Pope Francis I on Wednesday and said the choice of Cardinal Jorge Bergoglio of Argentina to ascend to the papacy speaks to the vitality of increasingly important Latin America.
Obama called the new pope "a champion of the poor and the most vulnerable among us."
"As the first pope from the Americas, his selection also speaks to the strength and vitality of a region that is increasingly shaping our world, and alongside millions of Hispanic Americans, those of us in the United States share the joy of this historic day," Obama said in a statement.