Mr Evans, 21, and Kate James, 20, want to move their son from Alder Hey Children's Hospital in Liverpool to Rome for treatment and have launched a second application to the Supreme Court.
They have already lost fights in the High Court, Court of Appeal, Supreme Court and European Court of Human Rights after Mr Justice Hayden ruled that doctors could stop providing life-support treatment for the 23-month old.
Judges have heard that Alfie, born on May 9 2016, is in a semi-vegetative state and has a degenerative neurological condition doctors have not definitively diagnosed.
Mr Justice Hayden said flying Alfie to a foreign hospital would be wrong and pointless.
Mr Evans posted on Facebook that he flew to Rome on Tuesday evening and met the Pope on Wednesday morning.
Posting photos of himself kissing Pope Francis's hand, he wrote: "Your holiness save our son."
During his general audience afterwards, the Pope led the crowd in a moment of silent prayer for Alfie.
At the beginning of April, Francis tweeted his support for the youngster, posting: "It is my sincere hope that everything necessary may be done in order to continue compassionately accompanying little Alfie Evans, and that the deep suffering of his parents may be heard.
"I am praying for Alfie, for his family and for all who are involved."
Earlier this week Francis said: "It's our duty to do everything to care for life."
