The pontiff said in a letter that woman would be able to seek absolution for having a termination during the Holy Year of Mercy which runs between December 8 2015 to November 20 2016.
Francis spoke of the "existential and moral ordeal" woman face once having had an abortion and said he had "met so many women who bear in their heart the scar of this agonizing and painful decision".
Currently Church teaching says abortion is a grave sin and those having had one should face automatic excommunication.
Peter Williams from Catholic Voices - the Church's mouthpiece in the UK - told Premier's News Hour it was "extraordinary".
He said previously the only person who could forgive a sin as serious as abortion was a bishop.
"It's quite a big move," he added.
"What it's trying to achieve is to emphasis to people the mercy of God the Father through the more easy access of the forgiveness that he offers.
"It's not a change in teaching at all, it's a change in discipline or pastoral practice.
"And so what this is saying is, right ok, rather than make it harder, make it easier and so making it easier will hopefully encourage many people who have gone through abortion to come back to confession".
The letter said: "The tragedy of abortion is experienced by some with a superficial awareness, as if not realizing the extreme harm that such an act entails.
"Many others, on the other hand, although experiencing this moment as a defeat, believe that they have no other option.
"I think in particular of all the women who have resorted to abortion. I am well aware of the pressure that has led them to this decision. I know that it is an existential and moral ordeal.
"I have met so many women who bear in their heart the scar of this agonizing and painful decision. What has happened is profoundly unjust; yet only understanding the truth of it can enable one not to lose hope.
"The forgiveness of God cannot be denied to one who has repented, especially when that person approaches the Sacrament of Confession with a sincere heart in order to obtain reconciliation with the Father.
"For this reason too, I have decided, notwithstanding anything to the contrary, to concede to all priests for the Jubilee Year the discretion to absolve of the sin of abortion those who have procured it and who, with contrite heart, seek forgiveness for it.
"May priests fulfil this great task by expressing words of genuine welcome combined with a reflection that explains the gravity of the sin committed, besides indicating a path of authentic conversion by which to obtain the true and generous forgiveness of the Father who renews all with his presence."
Peter Williams speaking to Premier's Hannah Tooley on the News Hour: