Cardinal Sean Brady made the appeal as he unveiled a statue of Father Patrick Peyton, known as the Rosary Priest.
"Make someone the gift of a prayer through text, twitter or e-mail every day," Times Online quoted Cardinal Brady as saying.
"Such a sea of prayer is sure to strengthen our sense of solidarity with one another and remind those who receive them that others really do care," he added.
The cardinal made the plea in the name of Father Peyton who worked tirelessly to promote the family recitation of the Rosary in the post-WWII years.
He continued: "In the name of Father Peyton I would like to make an appeal to every Christian in Ireland today who sends texts, twitters or uses e-mail.
"I appeal to you to think about setting up groups of prayer between you and your friends using these modern means of communication.
"I ask young people in particular to think of sending their friends and family an occasional twitter or text to say that you have prayed for them."
It was Father Peyton who had coined the saying: "The family that prays together, stays together."