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At the royal opening of the newly refurbished Dulwich Picture Gallery in south London this week, the Queen no doubt praised it as England's oldest public art gallery (it was completed in 1811). She may also have added that Sir John Soane's modest, windowless building in plebeian greyish-yellow stock brick, now grade II*-listed, still casts a much stronger spell on architects around the world than most grade I-listed buildings in her realm. Such landmark galleries as the Sainsbury Wing of London's National Gallery, designed by Robert Venturi, and Los Angeles' new Getty Museum by Richard Meyer have been influenced by Dulwich.

Soane's elegant stripped classicism is often seen as a precursor to modernism, as each gallery is capped by a generous lantern light that casts an even wash of natural daylight over the pictures hanging on the walls below. "As a display space, it's pretty wonderful, practically as well as aesthetically," enthuses the gallery's director, Desmond Shawe-Taylor. "What Soane didn't think about was visitor facilities," adds Shawe-Taylor. Hence the £9m project to add new annexes to house a café, toilets, an education studio and a lecture-cum-exhibition hall, as well as to restore Soane's original building.

Tampering with such an architectural shrine calls for a particularly sensitive approach. The competition-winning design by Rick Mather, masterplanner of London's South Bank Centre and architect of the Wallace Collection refurbishment appears, when seen from the main gate, to consist of nothing more than a flimsy pergola that runs around the border of a garden before disappearing into a new side entrance in the gallery.

The immediate effect is to tidy up the front garden: the "pergola" transforms the sprawling garden into a formal collegiate quadrangle (which, as Mather explains, was an unrealised ambition of Soane's), while masking the new accommodation behind it. This takes the form of single-storey spaces slotted into interstices in the site's layout.

On closer inspection, the delicate pergola turns out to be a substantial, fully enclosed cloister with glass walls and bronze roof. It provides visitors with sheltered access to the gallery, as well as the new-build elements – the exhibition-cum-lecture-hall by the main gates and the education studio and café fitted into a large recess in front of a neo-Jacobean school. Seen from the street, the exhibition hall presents a windowless brick wall that resembles the garden wall it replaces, albeit one-third higher.

Mather's original design for the pergola-like cloister was one of his trademark frameless all-glass structures. This was deemed too modern, so it was changed to a solid roof with ladder-like support columns, both in patinated bronze. Paradoxically, the solid roof makes the cloister less demonstrative, as the shadow it casts over the glass walls suppresses gleaming reflections. The supporting ladders have the filigree character of a Regency porch, which suits the garden setting.

The front facade of Soane's building has been restored, although not exactly to his original design. This was because, in 1912, two galleries were added either side of the central entrance in a style that did not match Soane's rhythm of blank arches. These extensions were considered too useful to demolish, but their front facades have been altered to mirror Soane's original arches, which had been covered up by the 1912 extensions. This approach may be politically incorrect in architectural conservation terms, but the effect is to make the facade look more acceptable than it did.

As for Soane's hallowed lantern lights, these have been replaced with what appear to be exact copies. Framed in timber, like the originals, the new lantern lights incorporate a sophisticated array of filters to protect the pictures from exposure to daylight. The glazing is made up of security glass on the outside and frosted glass on the inside, with louvred blinds sandwiched between them. In the vertical side panels, the louvred blinds adjust automatically to external daylight levels. In addition, blackout blinds extend across the entire lantern when the gallery is shut. The new lantern lights look and perform just as Soane intended, although with more sophisticated controls and with the inconspicuous addition of a few modern spotlights at their base and a couple of wires to support the black-out blinds.

Mather is no conservation architect, and his new extension is thoroughly contemporary in design. He has resisted the temptation to create a new building that shouts "look at me". Instead, he has craftily manipulated the site plan so as to lose the new accommodation in odd corners of the grounds. That is Mather's real achievement, and a really contemporary one, too.