In a speech to the European Parliament, the Argentinian pontiff called on the EU to help migrants fleeing the Middle East and Africa, saying it must not allow the Mediterranean to become a "vast cemetery."
The pope delivered a particularly strong issue on migrants, of whom more than 3,200 are estimated to have died this year trying to flee conflict in Syria, Iraq and Africa.
"We cannot allow the Mediterranean to become a vast cemetery!" he said.
"The boats landing daily on the shores of Europe are filled with men and women who need acceptance and assistance," he said.
It was the first papal visit to the French city since John-Paul II in 1988, but where his predecessor came at the end of the Cold War, Francis faces a more secular and eurosceptic continent riven by divisions and fears about the economy.
"Europe seems to give the impression of being somewhat elderly and haggard, feeling less and less a protagonist in a world which frequently regards it with aloofness," the 77-year-old pope told European lawmakers.
"We encounter a general impression of weariness and ageing, of a Europe which is now a 'grandmother', no longer fertile and vibrant."
He added: "The time has come for us to abandon the idea of a Europe which is fearful and self-absorbed, in order to revive and encourage a Europe of leadership."
Bells rang out from Catholic churches across Strasbourg to mark his visit, including the historic city-centre Cathedral, where hundreds of the faithful watched his speech on giant screens set up in front of the building.
But the crowd-loving Francis unusually left his popemobile behind and dedicated his time to addressing the European Parliament and the Council of Europe in a four-hour trip, the shortest abroad by any pope.
On his arrival at the huge glass and steel parliament building, Francis and European Parliament President Martin Schulz stood side by side for a flag-raising ceremony while a military band played the European anthem.
The leader of the world's 1.2 billion Roman Catholics was also meeting briefly with new European Commission chief Jean-Claude Juncker and outgoing European Council President Herman Van Rompuy.
Security was tight across Strasbourg with snipers posted on the top of buildings at the European Parliament and army sniffer dogs sent out to check the area before his arrival, AFP journalists said.
"It's going to be a big day," the pope, looking pale and tired but calm, told journalists on the plane before he arrived.
The climate has changed greatly since John-Paul II's visit and Vatican watchers say the pontiff must fight harder to be heard in an increasingly secular continent.
His speech repeated earlier warnings to Europe amid a rise in radicalisation, particularly among the disillusioned younger generation, and racism in countries hit hard by the economic crisis.
His visit has sparked protests in some quarters — including from a bare-breasted Femen rights group demonstrator who mounted the altar in Strasbourg cathedral on Monday — with critics angry over Schulz's decision to invite a religious leader to address a secular body.
Welcoming the pope, Schulz told parliament that his visit was important at a time when the economic crisis had caused a "tremendous loss of confidence in the European institutions."
Francis called for more to be done to tackle youth unemployment — which stands at an average 21.6 percent in the continent — and to care for those fleeing war zones and persecution, as well as the elderly.
He also spoke out on hot-button topics such as abortion and euthanasia — particularly after a slew of recent legislative changes in European countries.