In Catholicism, abortion is viewed as such a grave sin that it warrants automatic excommunication.
In most countries, only a bishop can approve forgiveness for abortion. They would then delegate an expert priest to hear the confession.
The change is only for the coming Jubilee Year, beginning in December.
However, the rule relaxation will not affect Catholics in England, Wales and Scotland as all priests there can already forgive abortion without seeking permission from a bishop.
The Pope said many women who sought an abortion did so because they "believe that they have no other option".
He added that he had "met so many women who bear in their heart the scar of this agonising and painful decision".
The easing of the rules is being seen as a shift in Catholic Church policy, reflecting the Pope's outspoken views on compassion and mercy.
"'I have decided, notwithstanding anything to the contrary, to allow all priests for the Jubilee Year to absolve of the sin of abortion those who procure it and who also seek forgiveness," Pope Francis said.
Pope Francis is aware that the decision will not be welcomed by traditionalists, says the BBC's David Willey.
However, in contrast to his predecessors, he regards compassion and mercy as virtues that outclass all others.
The Jubilee Year runs from 8 December 2015 to 26 November 2016 and is seen as one of the Catholic Church's most important events.
It normally takes place every 25 years unless a pope declares an extraordinary one to draw attention to a particular issue.
The Pope has previously denounced abortion as part of a throw-away culture.
"It is horrific even to think that there are children, victims of abortion, who will never see the light of day," he said in 2014.
"Unfortunately, what is thrown away is not only food and dispensable objects, but often human beings themselves, who are discarded as unnecessary," he added.
Pope Francis has been viewed as progressive on other controversial issues. In 2013, on the topic of homosexuality, he said: "Who am I to judge?"
However, he has not changed Church policy on contraception.
Speaking earlier this year, Pope Francis said that although contraception remains forbidden, there was no need for people to breed "like rabbits" to be good Catholics.

Laser mapping will be used to measure the impact of extreme weather on six Welsh islands in the Irish sea.
The aircraft beams will assess the effect of rising sea levels and storms on locations, including Bardsey and Ramsey islands.
The mapping will also create "stunning" animations as part of the first 3D data for many of the islands.
Led by the Royal Commission of the Ancient and Historical Monuments of Wales, the five-year project will assess the damage done to archaeology and heritage sites by coastal erosion.
"This is an exciting new project, bringing a strong partnership of archaeologists, geoscientists and maritime specialists to bear on the significant challenges posed by climate change," said commission secretary Christopher Catling.

Lee Radford's side crossed eight times as they thrashed Salford at home in their opening match of the season.
Fetuli Talanoa touched down twice in France, adding to scores from Sika Manu, Josh Bowden, Jack Logan, Jamie Shaul and captain Gareth Ellis.
Winger Pat Richards scored all 10 of Catalans' points in the first half.
Media playback is not supported on this device
The Dragons have lost their first two games of the campaign and such a heavy home defeat early in the season will be a concern, with much of their success in recent years being based on a formidable home record.
Hull top the table ahead of Wigan, with both sides starting the campaign with two victories.
Former England international Ellis was making his first appearance for the Black and Whites since July after recovering from a serious Achilles injury.
Catalans Dragons: Gigot; Yaha, Inu, Duport, Richards; Carney, Myler; Anderson, Pelissier, Bousquet, Stewart, Horo, Baitieri.
Replacements: Taylor, Mounis, Bosc, Elima.
Hull FC: Shaul; Michaels, Logan, Tuimavave, Talanoa; Pryce, Sneyd; Taylor, Houghton, Watts, Minichiello, Manu, Thompson.
Replacements: Pritchard, Ellis, Bowden, Washbrook.
Referee: Robert Hicks
Pope Francis is making it easier for women and doctors to seek forgiveness for abortion, by allowing all priests to forgive it.
[NEXT_CONCEPT]
Animation showing the threat of climate change is on the horizon, thanks to new 3D laser mapping.
[NEXT_CONCEPT]
Hull FC secured a second convincing Super League win in succession, scoring seven tries in a stunning victory against Catalans Dragons in Perpignan.