Bishops who are "negligent" in dealing with priests committing abuse will be removed under the new legal procedures.
The decree comes in response to long-running demands by abuse victims and their supporters to hold bishops accountable if they fail to protect their flocks from paedophiles.
Existing laws relating to abuse cases would be tightened, the Pope said.
He acknowledged that canon law already allows for a bishop to be removed for negligence but says he wants a more precise definition of the "grave reasons" that could lead to dismissal.
"I intend to specify that among these so-called 'serious reasons' is the negligence of bishops in the exercise of their functions, especially in cases of sexual abuse of minors or vulnerable adults," Pope Francis wrote.
Vatican spokesman Federico Lombardi was quoted by the AFP news agency as saying that the pontiff had also established a group of lawyers to help him make decisions that could result in the dismissal of a bishop.
His motu proprio (on his own impulse) decree emphasises that the Church "loves all its sons, but cares for and protects with special attention those who are weakest and defenceless".
This explains the requirement for priests, especially bishops, to display "particular diligence" in this area, the Pope said.
Some bishops have covered up abuse by transferring perpetrators from parish to parish rather than reporting them to police.
Pope Francis set up a Vatican commission to establish best practice in relation to abuse cases and expose wrongdoing in parishes in 2014.
The pontiff tweeted on Saturday: "Let us hear the cry of the victims and those suffering, no family without a home, no child without a childhood."
The Roman Catholic Church has for much of the last 15 years been forced on the defensive by scandals involving priests who are alleged to have abused children and then been transferred rather than handed over to the authorities.
Pope Francis has approved measures to sack bishops who mishandle child sexual abuse cases, a papal decree says.