During the 25-minute meeting in the Apostolic Palace, Francis "implored anew God's forgiveness for the sins and failings of the church and its members," the Vatican said.
He "expressed the desire that this humble recognition of the failings of that period, which unfortunately disfigured the face of the church, may contribute to a 'purification of memory' and may promote, in hope and renewed trust, a future of peace."
The Rwandan government has long pressured the church to apologize for its complicity in the genocide, but both the Vatican and the local church have been reluctant to do so. The church has long said those church officials who committed crimes acted individually.
In 1996, St. John Paul II refused to take blame on the church's part for what transpired in Rwanda, saying in a letter to Rwandan bishops that: "The church in itself cannot be held responsible for the misdeeds of its members who have acted against evangelical law." Four years later, however, he did make a general apology for a host of Catholic sins and crimes over its 2,000-year history.
Amid continued pressure from the government, Rwanda's Catholic bishops last year apologized for "all the wrongs the church committed."
The ministry of local government rejected the apology then as inadequate. During Rwanda's annual dialogue in December, Kagame said he didn't understand why the church was so reluctant to apologize for genocide when popes have apologized for much lesser crimes.
"I don't understand why the pope would apologize for sexual offenses, whether it is in the U.S., Ireland or Australia, but cannot apologize for the role of the church in the genocide that happened here," Kagame said at the time.
On Monday, the Rwandan government called Francis' meeting with Kagame a "positive step forward."
"Today's meeting was characterized by a spirit of openness and mutual respect," said Foreign Minister Louise Mushikiwabo. "It allows us to build a stronger base for restoring harmony between Rwandans and the Catholic Church."
However, Mushikiwabo repeated charges that even before 1994, Catholic institutions helped divide Rwandans and "laid the intellectual foundation for genocide ideology."
"Today, genocide denial and trivialization continue to flourish in certain groups within the church and genocide suspects have been shielded from justice within Catholic institutions," the statement said.