The continent's people "are tempted to yield to our own selfish interests and to consider putting up fences", he said.
The pontiff was awarded the International Charlemagne Prize, for his "message of hope and encouragement".
During his acceptance speech in front of German chancellor Angela Merkel, Italian premier Matteo Renzi and Spain's King Felipe VI he said: "I dream of a Europe where being a migrant is not a crime, but a summons to a greater commitment on behalf of the dignity of every human being.
"I dream of a Europe that promotes and protects the rights of everyone, without neglecting its duties toward all. I dream of a Europe of which it will not be said that its commitment to human rights was its last utopia."
Echoing the famous "I have a dream" speech by US civil rights leader Martin Luther King Jr, Francis offered his vision of a Europe that cares for children, the elderly, the poor and the infirm, as well as "those newcomers seeking acceptance because they have lost everything and need shelter".
He also said youth unemployment was sapping the continent of its dynamism, and he called for new economic models that are "more inclusive and equitable".
"There is an impression that Europe is declining, that it has lost its ability to be innovative and creative, that it is more concerned with preserving and dominating spaces than with generating processes of inclusions and change," Francis said.
"A memory transfusion can free us from today's temptation to build hastily on the shifting stands of immediate results, which may produce quick and easy short-term political gains, but do not enhance human fulfilment," he said.
Pope Francis also met with the EU Commission president Jean-Claude Junker at the ceremony.
Mr Junker told him: "When you take in 12 refugees, in proportion to the population of the Vatican that is more than any EU member state - you fill our hearts with new courage."
The Charlemagne prize, consisting of a medal and a citation, is awarded annually by the German city of Aachen for contributions to European unity.
Previous winners include former US president Bill Clinton and St John Paul II, who received a special edition of the prize in 2004.