Dozens of Catholic priests are believed to have been complicit in the killings of some 800,000 people by Hutu extremists in 1994.
Several massacres were carried out in churches where people sought sanctuary.
The Vatican has, until now, maintained that the Church as an institution bore no responsibility.
On Monday, Pope Francis conveyed his "profound sadness" for the "genocide against the Tutsi," the Vatican said in a statement.
He begged for God's forgiveness "for the sins and failings of the Church and its members" who, the statement said, had "succumbed to hatred and violence".
In November, the government in Kigali requested an apology from the Vatican for the role that some Catholic priests and religious figures played in the massacres of ethnic Tutsis and moderate Hutus.
"Today, genocide denial and trivialisation continue to flourish in certain groups within the Church and genocide suspects have been shielded from justice within Catholic institutions," a government statement said earlier on Monday.
The Pope's statement, which he said he hoped would help Rwanda heal, followed a meeting in Rome with Rwandan President Paul Kagame.
Mr Kagame described the comments on Monday as a great moment and a new chapter in relations between his country and the Vatican.
Mr Kagame, Tutsi, led a rebel force to halt the slaughter in 1994 as accusations surfaced that some priests and nuns had taken part in the killings.
On 6 April 1994, a plane carrying then-President Juvenal Habyarimana - a Hutu - was shot down, killing everyone on board.
Hutu extremists blamed the Tutsi rebel group the Rwandan Patriotic Front (RPF), and immediately started a well-organised campaign of slaughter.
The RPF said the plane had been shot down by Hutus to provide an excuse for the genocide.
Between April and June, an estimated 800,000 people - mostly ethnic Tutsis and moderate Hutus - died at the hands of Hutu extremists.
Pope Francis has asked for "God's forgiveness" for the "sins and failings" of the Roman Catholic Church at the time of the Rwandan genocide.