Lee, 80, was accompanied to the audience in Saint Peter's Square by her daughter and Coogan, the British actor who co-stars in and produced the film.
Philomena, which has Judi Dench in the starring role, was warmly received at the last Venice film festival and is in the running for the Best Picture award at this year's Oscars.
It tells how, in 1952, as a pregnant teenager seen as a "fallen woman" in deeply Catholic Ireland, she went to a convent to give birth to her son Anthony, who was then taken to be adopted by an American couple.
After keeping her secret for 50 years, Lee decides to track down Anthony, helped by a former BBC journalist, played in the film by Coogan.
"I hope and I believe that Pope Francis will join with me in my struggle to help thousands of mothers and children put the words 'The End' over their painful past," Lee said, according to Italy's ANSA news agency.
Today, Lee works for the Philomena Project, which tries to help other mothers find their children who were taken away, notably by asking Ireland to pass a law unsealing adoption records.
There was little chance of the pontiff seeing the movie of Lee's struggle, though. A Vatican spokesman, Father Federico Lombardi, said: "The Pope does not watch films."
Watch a trailer of the film: