The pardon was initially a temporary measure introduced during the Catholic Church's Holy Year of Mercy, which ended on Sunday.
Until then, granting absolution for an abortion was a right reserved for bishops or a designated chief confessor of a diocese in most parts of the world.
Now, in a document known as an apostolic letter following the end of the Holy Year of Mercy, Pope Francis has announced an indefinite extension of priests' right to do so.
But the Pope has stressed the Church still considers abortion a grave sin.
"I thought about the habit of getting rid of babies before they are born, this horrendous crime. They get rid of them because it is better like that, because it is convenient, it is a huge responsibility. That is a grave sin, no? A huge responsibility."
In the letter made public by the Vatican, he also wrote, "There is no sin that God's mercy cannot reach and wipe away when it finds a repentant heart seeking to be reconciled" with God.
The move is believed to be part of the Pope's work to make the Church more inclusive and forgiving, something which has become characteristic of his papacy.
In Church teaching, abortion is such a serious sin that those who procure or perform it incur an automatic excommunication until absolved in confession.
But in the Vatican, Archbishop Rino Fisichella explains rank-and-file priests can now forgive women who have abortions, as well as the doctors who perform them.
"Technically, everyone who is involved in an abortion. That is, the woman, nurses, the doctor, whoever supports this matter. The sin of abortion involves all. Thus, the pardon of the sin of abortion equally covers everyone, involves everyone who has played a part in this sin."
British-based abortion-rights advocate Marge Berer has told Al Jazeera television she cautiously welcomes the move.
"I think it was always very cruel to excommunicate women who'd had an abortion who actually confessed it. But my impression is that most Catholic women don't confess it. And our feeling, as an abortion-rights campaign, is that women shouldn't need forgiveness for abortion. An abortion shouldn't be treated either as a sin or a crime."
She says she hopes the move is just the beginning of change in the Church.
"The Church is letting women down in many ways, and it would be nice if they would consider changing their position at some point. And I think that this Pope, maybe ... It was started at the beginning when he first came on board that he was going to be a little bit softer on the subject. But it's not just his decision either. There's a whole hierarchy in the Church who can make these decisions and influence him."
