In an exclusive interview with Reuters, Cardinal Sandri, 69, an Argentine, also said the next pope should not be chosen according to a geographic area but must be a "saintly man" qualified to lead the Church in a time of crisis.
He said one of the greatest challenges facing the Church was trying to win back those suffering from a "loss of faith" who had "turned their back on God" and the Church of their fathers.
Sandri, an experienced diplomat and past number two in the Vatican bureaucracy, is expected to wield great influence in the choice of the man to succeed Pope Emeritus Benedict XVI.
"The role of women in the world has increased and this is something the Church has to ask itself about," Sandri said in his office just outside St Peter's Square where he heads the Vatican department for Eastern Catholic Churches.
"They must have a much more important role in the life of the Church ... so that they can contribute to Church life in so many areas which are now, in part, open only to men ... This will be a challenge for us in the future."
Currently only two women are under-secretaries, one a nun and one a lay woman.
Sandri, a son of Italian immigrants who has been mentioned as a papal candidate, said it was "only right" that women should have more key positions in the Vatican administration "where they can make a very important contribution because of their qualifications".
He added: "But they must also be co-participants in the dialogue and the analysis of the life of the Church and in (other) areas, even in the formation of priests, where they can play a very, very important role."
The Church teaches that women cannot become priests because Jesus willingly chose only men as his apostles. Sandri gave no indication that the rule could be changed.
Sandri spoke as cardinals from around the world gathered for preliminary meetings on Monday ahead of a closed-door conclave to choose a new pope which is expected to begin in the Sistine Chapel around March 10.
He said the next pope should have holiness, physical strength, communications skills, and the ability to govern.
"The problem is finding the four qualities together. Sometimes someone has an excess of one quality and less of others ... for me, the most important thing is that he be a man of faith who is not afraid," the cardinal said.
"I would like to see the new pope be someone who above all is, if not yet a saint - that is difficult to ask for - someone who is at least walking towards sanctity through a life of humility, work, prayer and witness to the Gospel," he said.
There has been much speculation that the next pope could be a non-European, perhaps from Africa or Asia, both places where the Church is more vibrant and less ossified than in Europe and in other developed areas.
But Sandri warned against picking a pope by his passport.
"I am skeptical about relying on geographical definitions. Someone should not be elected pope just because he is not a European," the cardinal said.
Cardinal Peter Turkson, 64, from Ghana, who heads the Vatican's justice and peace department, is seen as a leading African candidate.
"The Church is ready for a black pope but maybe the world is not," Sandri said. "We are open to anyone as long he is the best prepared, the best qualified, to face a time that is so difficult for the Church and the world."