As the world's biggest, oldest, most influential and perhaps most colourful institution of any kind, the Catholic church has surely merited more attention than cinema has accorded it. Angels & Demons and Habemus Papam gave a hint of the possibilities, and that somewhat minor branch of pastoral activity, exorcism, has been more than adequately explored. Otherwise, we've had saintly but boring priests such as those of The Bells of St Mary's and Angels with Dirty Faces or absurdly delightful nuns like those in The Sound of Music and The Nun's Story.

In part, the prevalence of such sympathetic portrayals has reflected the lobbying power of the Legion of Decency, an extraordinarily effective body set up in the 1930s by Catholic bishops to bring Hollywood to heel. You might have thought that attitudes would have been turned upside down by the paedophile priests scandal, but things have changed less than you'd expect.

Last year, a documentary, Mea Maxima Culpa: Silence in the House of God, provided a cogent account both of the abuse inflicted and of the coverup that followed. Priests and nuns have been presented more acerbically than before in films such as Doubt; all the same, a definitive big-screen takedown has so far failed to appear. Yet surely nothing could have damned the church more decisively than the sexual exploitation of children entrusted to it by their parents.

Well, maybe there's one thing that could. How about stealing babies from vulnerable teenagers, selling them overseas and then preventing them tracing their parents by burning records of the transactions? This is the behaviour, as practised in real life by Irish nuns assigned to the care of fallen women, that's addressed by Stephen Frears in Philomena. Such a film, we might reasonably imagine, ought to do for the Vatican what Ewan McGregor's Camerlengo Patrick McKenna so nearly achieved with his antimatter bomb in Angels & Demons. (Spoilers ahead.)

Philomena's hero, Steve Coogan's Martin Sixsmith, looks like the man for the job. He knows the territory, as he's a Catholic himself: he may be lapsed (as is Coogan), but the institution to which he was consigned in infancy provides no exit procedure. As a former BBC journalist and Blairite spin-doctor, he embodies the godless cynicism and presumptuous sanctimony of his metropolitan tribe. The sin that brought about Philomena's plight arouses only his uncomprehending contempt: "Why would God bestow upon us a sexual desire he expects us to resist?"

Martin's confederate looks even better placed to bring about Rome's comeuppance. Judi Dench's Philomena has spent a lifetime nursing her grief for the child snatched from her by a wimpled witch. Now, thanks to the secular miracle of Martin's Googling, she's going to find out just what happened to her son; then she'll return to confront her cloistered tormenter of old. Cue, surely, a hell-hath-no-fury climax that will put to shame Carrie, Kill List, Medea, Enough, Lipstick, She-Devil and I Spit on Your Grave.

But no. It's Martin who's consumed by righteous wrath; Philomena bestows her forgiveness on the now aged but still unrepentant author of her lifelong pain. More puzzlingly still, she reaffirms her unquestioning allegiance to the religion that's provided both the means of and justification for her maltreatment. She and her persecutor emerge united in subjection to something greater than their own ugly story and mysteriously untarnished by it: their church.

Martin responds with a look of dumbfounded, resentful fury, the like of which we haven't seen since that blinkered TV exec rejected monkey tennis. His reaction reflects a wider phenomenon — the bafflement of the enemies of Catholicism over its ability to rise above what should surely have been the terminal scandals of recent decades. Apparently, the church of Rome can molest altar boys and peddle abducted infants as much as it likes; today, replete with its bling bishops and pernicious social edicts, it still enjoys the devotion of over a billion satisfied adherents in places from Manila to Montreal and Madrid to Montevideo.

The story of Jesus requires all Christians to believe at least six impossible things before breakfast; Catholics, however, must also accept that the son of God saw fit to entrust an institution that would be endemically beset by corruption and cruelty with the dissemination of his Gospel, the administration of his sacraments and the exercise of his charity. To atheist, rationalist Benthamites such as Martin it just doesn't make sense. Philomena shows why it does.

At the end of the film, it's Martin who's bitter and confounded. Philomena, for all that she's been through, is both cheery and serene. Such is the priceless reward that only her faith can yield. How she managed to cling to it while it slipped from Martin's grasp remains beyond his understanding. Yet her ingenuousness turns out to have been more productive than his scepticism. The Catholic church survives its scandals, Philomena's story shows us, because it delivers the goods.