There has been a vogue recently for encounters between past and present masters. In Madrid, Paris and London, Picasso has been set in the context of influences such as Velázquez, Delacroix and Manet. Less successfully, Jan Fabre, the Belgian contemporary artist was let loose in the Richelieu wing of the Louvre: displayed alongside Rembrandt and Rubens, Fabre’s flimsy installations were grist to the mill of traditionalists who swear no contemporary artist can match up to the old masters.

The decision to confront Francis Bacon with Caravaggio is also fraught with risk. As the curators acknowledge, the 17th-century master exerted no direct influence on the Irish-born modernist. (Indeed, Bacon, who revered Velázquez, Rembrandt and Picasso, was influenced by Poussin, one of Caravaggio’s harshest critics.) What the pair do share is a revolutionary approach to the human figure, a fascination with anatomy, and a vision that is simultaneously sacred and profane. Both have been tagged as icons of gay, tormented genius whose decadent and violent lives – Bacon’s lover committed suicide; Caravaggio killed a man and wounded several others – fuelled the anguish in their canvases.

Yet it is the differences between them that make this show compelling. Caravaggio was a Catholic; Bacon an avowed atheist. Caravaggio was the flag-bearer of the Counter-Reformation, charged with seducing the faithful away from Lutheran temptation. In Bacon’s age, secularity was yesterday’s news and artists painted to please themselves. Bacon’s refusal to relinquish the human figure while artists all around him turned to abstraction looked quasi-reactionary. Surrounded by the distorted idealism of high mannerism, Caravaggio’s fidelity to the real – he transcribed every wrinkle, every hair, every frantic gesture – saw him pilloried as a radical.

Caravaggio’s ‘Conversion on the Road to Damascus’ (1601)

At first, the encounter jars. Built around the ancient-to-baroque art collection amassed by the 17th-century Roman cardinal Scipione Borghese, the Galleria Borghese is one of the finest small museums in the world. But Bacon’s triptychs, “August” (1972) and “Triptych inspired by the Oresteia of Aeschylus” (1981), are eclipsed by the combination of two mighty Caravaggio canvases, “The Conversion on the Road to Damascus” (1601) and “The Resurrection of Lazarus” (1609), and the jaw-dropping lavishness of the entrance hall, with its frescoes, statues and mosaic floors.

The display is partly to blame. Mounting the triptychs with their colour-field backgrounds against temporary panels in a pink that tones not only with Bacon’s own colours but also the ceiling fresco is a perilous reminder of Bacon’s original trade as an interior decorator.

But the real problem is the temporal leap demanded of the eye. Plunging a 20th-century artist with a predilection for deconstructing the human figure into a room devoted to classical beauty is an optical challenge. Painted in oil devoid of the sensuous impasto and tenebrous chiaroscuro that make premodern art so seductive, Bacon’s figures look like cartoons: cheeky, graphic teases cocking a snook at the grand old patriarch whose transcendent beauty puts him beyond threat.

After a few minutes, however, these visual hurdles recede and one becomes aware that a gripping dialogue across centuries and belief systems is taking place. What is at stake here is faith. Although he had a complex relationship with religion, repeatedly painting crucifixions, Popes, and triptychs, the imagery of “August” tells us that Bacon’s world was a redemption-free zone.

Francis Bacon’s ‘Study of George Dyer’ (1969)

One of a cycle of “black triptychs” painted after the suicide of his lover George Dyer in a Paris hotel room, the work depicts two semi-naked male figures, leaking out their life force into sinister flesh-pink puddles. Flanked by this pair, a nebulous spillage in mauve, grey and white is Bacon’s chilly vision of a sexual coupling. These abject scenes are framed by a trio of black portals whose matt, merciless, impenetrable surfaces suggest nothing lies on the other side.

Caravaggio’s painting of Lazarus makes a powerful case for the alternative. As a divine glow plays across the scene – the nude torso of the beggar, Christ’s omnipotent, out-stretched arm – the cavern’s tawny-lit, deliquescent darkness truly seems the territory of miracles.

Other than Piero della Francesca, no artist knew better than Caravaggio that light was the Catholic painter’s greatest ally. In the “Conversion of St Paul”, a breathtaking image of Paul prone beneath the raised hoof of a piebald horse, he floods a dynamic, diaphanous glow on to the horse’s silky hide, making the animal the hero of the painting and reminding us that we are all equal under the eyes of God.

The contest is more equal in the room devoted to portraits. Here, two early Caravaggio canvases, “Young Man with a Basket of Fruit” (1593-1595) and “Self-portrait as Bacchus” (1593-1595), show Caravaggio developing the style that would ensure his lasting fame. Although often presented as “the first modern painter” for his refusal to idealise nature, the Lombard-born artist was steeped in the lessons of classical antiquity. Thus he renders every bloom, vein and blemish on his fruit basket in Flemish-style detail yet the boy who holds it, with his purple-shadowed throat and parted, rosy lips, possesses the sculpted perfection of a Michelangelo.

Bacon’s contribution is anchored by “Head VI” (1949),  one of more than 50 paintings based on Velázquez’s portrait of Pope Innocent X. Bacon was no draughtsman but Velázquez’s outline bestows a steely gravitas that is the perfect counterpoint to Bacon’s disfigurement. By dragging paint across an untreated canvas and adding that fathomless, shrieking mouth, Bacon creates an expression of such archetypal horror you sense all his demons – death, faith, masculinity, patriarchal authority – distilled into that single image.

Of course, these were Caravaggio’s demons too. In the the show’s finale, a clutch of marvellous works by the Italian – “Madonna di Loreto” (1604-1605), “St Jerome” (1605-1606), “Madonna dei Palafrenieri” (1605-1606) – include his “David with the Head of Goliath” (1610). One of Caravaggio’s final paintings, it’s said that the Philistine’s bloodless, open-mouthed visage is a self-portrait of the artist at the end of his life, when he was tortured by guilt and by the thought of his own mortality.

Bacon, who once described his crucifixion scenes as self-portraits, would have understood. A weakness of this show is that it barely contains any of the canvases – the Guggenheim’s “Three Studies for a Crucifixion” (1962), for example – where the Irishman explodes the human body into viscous, blood-hued rubbles of flesh. Only one, “Study from Portrait of Pope Innocent X” (1953), the Pope’s face dissolving into a garnet-red froth to match his robe, hints at the profound sense of revulsion – for death, self, the human condition – that animates his most powerful work.

Ultimately, the absence of a few masterpieces doesn’t matter. Although the curator Anna Coliva writes that the show is “not an exhibition of history”, that’s exactly what it is. But the lesson is made thrilling by the aesthetic power of the works.

Leaving Caravaggio and Bacon aside, the museum offers a whistle-stop tour from ancient art to Rubens, by way of Bellini, Raphael and Correggio. In this company, we see how Caravaggio’s determination to paint the poor, the old and the ugly heralded the slow disintegration of classical ideals of beauty. With modernity came the revelation that man was no longer “the measure of all things”. Bacon’s sorry, disfigured souls mark the final act of the tragedy. Perhaps he was lucky. If he had been born 50 years later he might, like Jan Fabre, be making art out of beetles.

‘Caravaggio Bacon’, Galleria Borghese, Rome, until January 24 2010; www.galleriaborghese.it