Among the Italian Renaissance highlights of the Musée Jacquemart-André, a collection open to the public since 1913, is a Madonna and Child. It was painted, c.1470, in tempera on wood by the young artist from Città del Pieve, Pietro Vannucci (c.1450-1523), who is nicknamed Perugino after the city of Perugia, where he mainly worked and where, indeed, he died.

Nélie Jacquemart acquired the painting in 1884 at a sale in Rome, when it was attributed to Antonio Pollaiuolo, recognising its indebtedness to the Flemish mastery of oil painting which flourished at the time in Florence. It was not until 1932 that it was given to Perugino, an attestation that is more than amply demonstrated in this fascinating show mounted for the Institut de France.

The exhibition is staged for the museum - which houses possibly the finest private art collection in Paris - by Hubert le Gall, who first showed the Altenburg Collection of Italian Primitives here in 2009. A dozen subsequent installations have included Fra Angelico (2011), and later artists such as the Caillebotte brothers, Eugène Boudin (2013), and most recently Watteau and Fragonard.

The exhibition was originally planned to be just of Perugino, and the temptation to add in a more publicly recognised name to the title buys into a continuing scholarly problem. The question whether Raphael (1483-1520), as Giorgio Vasari in his Lives of the Artists would have us believe, ever could have been Perugino's pupil comes at the end of a show that features more than 50 works of art that showcase Perugino in a range of painting styles.

But, to begin at the outset, the first room delineates Perugino's own prodigious skill. Side by side (for the first time since more than 100 years ago, when fragments of a Marian altarpiece, thought to have been housed in the Florentine Basilica della Santissima Annunziata but long since lost, were sold off) are two panels from English collections. They are reckoned to date to the artist's early twenties, when he had been a few years in Perugia, where he was well established after his apprenticeship. He was a member of the prestigious Guild of St Luke by 1472.

In one, The Birth of the Virgin (Walker Art Gallery, Liverpool), St Anne struggles to sit up in bed, readying herself to receive an inevitable stream of visitors. In the other, the Vision of Pope Liberius and the Virgin of the Snows (Polesden Lacey, National Trust), wealthy patricians greet the pope in an idealised city scene of antique Rome. The scale of the works (just 20 × 40cm) suggests that there were three other panels for the missing altarpiece. Where are they now?

By way of contrast, the panel of Saints Anthony of Padua and Sebastian (Nantes) looks back to Perugino's teacher, Bartolomeo Caporali, 30 years his senior. The ornate gold background and formal dress of the saints harks back to an earlier artificial style that was already becoming unfashionably out of date, even though it had served Caporali well, as his Madonna and Child from the house of the Poor Clares at Santa Maria di Monteluce (1465) shows vividly in the next room.

If we want evidence of how Perugino was a master of several styles as early as the mid-1470s, we have only to look at the fresco of two saints invoked against the plague. When pestilence ravaged the nearby Umbrian city of Deruta in 1476, the city fathers proclaimed that the August feast days of St Romanus and St Roch henceforward be observed as obligatory feasts. Perugino frescoed both invocatory saints for the left-hand wall of the Franciscan church. He included a topographically accurate cityscape at their feet. It is dominated by the Gothic campanile of the Franciscan church itself in the centre. The city gates and the Church of Santa Maria dei Consoli are all immediately recognisable.

Whereas the pair of saints in Nantes looked back to a formal realm of candlelit churches burnished with glinting gold, the commission at Deruta offered the chance of a much freer handling of the figures. The perspective is already much more realistic, and suggests a degree of movement which is wholly new in the depiction of holy men and women within churches, where the image had to be prayed in front of, and was not simply decoration.

Even the paintings of the Madonna and Child are looser, and offer a sweetness and elegance that are at odds with the stiffer and more reserved representations of the Renaissance, which had continued to look back to the icons of the Eastern Church. In this, Perugino was not alone, as we see from a tempera panel by his older contemporary Sandro Botticelli (1445-1510).

It is generous indeed that we should here see, side by side, paintings from the Borghese collection in Rome; London, with St John the Baptist (National Gallery); and the Jacquemart, already mentioned (c.1470s), in which the Madonna's downcast eyes, although notionally on her open prayer book, suggest that she is already contemplating the Passion, as her son holds a string attached to a symbolic chaffinch's leg.

The popular success of such paintings, together with the spread of Perugino's undoubted reputation, led to his summons to Rome to paint for Pope Sixtus IV in 1479. Although the Chapel of the Conception at St Peter's in the Vatican has since been redeveloped, it was here that he worked with other painters from Florence, such as Botticelli and Cosimo di Lorenzo Rosselli (1439-1507). Several of his large panels still survive on the walls of the Sistine Chapel itself.

Leaving Rome in 1482, the Umbrian master returned north, and certainly visited Venice (possibly twice) in the following decade. The lucidity and translucence of light in his pictures at this time becomes increasingly evident. An obvious shift can be seen in the Madonna and Child from Washington, DC (1496), which is likely to be one of the most loved pictures in the show, and in the Madonna and Child with Members of the Confraternity of Santa Maria Novella, commissioned in the same year for their chapel at Porta Sant' Angelo, in Perugia.

The elegantly drawn Penitence of St Jerome (Royal Collection) was presumably painted originally for a processional banner. When it was presented to Prince Albert in 1846, he had it hung in his bedroom at Osborne House, which might well illustrate A. N. Wilson's recent thesis about the younger Victoria's marital relations.

Surprisingly, the languorous St Sebastian, which has been in the Borghese Gallery in Rome since at least 1650, next to it has often been dismissed as a copy or the work of a pupil. Professor Tom Henry is surely right to defend its outstanding authenticity. Besides being a saint who has been invoked in times of pestilence, Sebastian was often portrayed as an alter Christus. In this composition, the parallel with "Christ at the column" is so evident that the image is devotionally demanding.

An indication of the Umbrian master's post-Venetian style is shown in the recently published double images of the Virgin and the Christ crowned with thorns which is in a private Swiss collection. These two small panels (each 33 × 27cm) echo the enigmatic faces that we associate with Antonello da Messina. The pictures are bound in embossed-leather covers, much as an ancient codex that can be glimpsed in the mirror supporting them, and would have served as a diptych.

The last two rooms in the exhibition not only introduce us to the precocious Raphael, but seek to address the conundrum of the artistic relationship that Vasari reported was that of master and pupil. Against this is the unlikelihood that an 11-year-old who had just lost his father would be sent off to work with a foreign painter.

One above the other are panels that formed the predella scenes beneath two very different altarpieces, one loaned from the Vatican (The Oddi Retable of 1504/05), and the other from Fano, where it was beneath an altarpiece, commissioned in 1488 and dated 1497, making it surely too early for Raphael to be involved in the works composition.

The handling in both is very different; the undoubted Raphael, painted for San Francesco al Prato in Perugia, simplifies the overall design, stripping the architectural scene bare, and concentrating on one or two points in each. In his narrative, the annunciation takes place in a bare hall with undecorated columns, and the urn-like altar for the presentation is the only fussy detail that might distract us.

Perugino outlived the much younger Raphael, and, in turn, had become influenced by the younger artist, whose reputation in papal Rome had overtaken him. This is brilliantly shown in the monumentality of the figures of the Apostle Philip and the Latin Doctor St Augustine of Hippo. One of possibly 30 panels for a high-altarpiece for the Augustinians in Perugia (now in Toulouse), the two figures (dated 1502-12) combine the grace and eloquence characteristic of Perugino even in the later years of his life. 

"Pietro Perugino: Master of Raphael" runs at the Musée Jacquemart-André, 158 boulevard Haussmann, Paris 75008, until 19 January. It is open daily from 10 a.m. to 6 p.m. (Mondays to 9 p.m.). Phone 00 33 1 45 62 11 59.

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