Anyone attending a performance at La Scala often comes away feeling lucky. It has a world class opera orchestra that is hard to beat in the Italian repertoire. The same players make up a thoroughly decent symphony orchestra, which attracts some glittering names from the conductor world to direct them. Add to that the smattering of top-league bands and their superhuman directors that on occasion come this way, and good fortune quickly starts to feel like blessedness.

The latest arrival is Mariss Jansons and his BRSO, last night pulling into the final leg of its whistle-stop European tour. For how much longer will we get to hear this mighty conductor-orchestra combination? Jansons, now 73, has defied health problems for a number years, but finally tied up his commitments with The Royal Concertgebouw Orchestra (where he was simultaneously Chief Conductor) at the end of last year.

Paperwork alone would suggest that his relationship with the BRSO will outlive that with the RCO by some margin. Jansons recently extended his contract with the orchestra to 2021, citing his long-held ambition to secure a new Munich concert hall as a key motivation. That ambition was achieved at the end of last year, but Jansons is nevertheless staying on to bed them in. Anyone within spitting distance of an upcoming concert should rush to buy a ticket now.

Tonight's choice of repertoire was precision calculated to demonstrate this orchestra's strengths. Shostakovich's "Leningrad" Symphony both roars with the horrors of a city under siege and has long, rambling passages where the orchestra would demonstrate its mastery of balance and blend. Jansons is a passionate advocate for this music. He gave up conducting Russian repertoire in the early 2000s by way of a change, but made an exception for Shostakovich, who was too special to let go.

We get our first experience of these forces at full pelt at the pinnacle of the first movement's creeping, monstrous march. It's a sound that makes you forget to breathe: deep, monolithic and with a high-definition sound in which every component gleams. The buildup had been measured – pinpoint snares and lovingly sculpted trumpets – but Jansons applies the throttle when the music shifts gear. From 0 to 100 in a split second, we were assaulted with wave after wave of shocking rage.

"It's like driving a Rolls Royce," suggested Jansons in a recent interview for Italy's Amadeus magazine. Being a radio orchestra is apparently key to its success. All of its performances are broadcast or recorded, so technical precision becomes the basic expectation. Rehearsals are used to build on this foundation, with time dedicated largely to interpretative matters.

One element Jansons has clearly cultivated is cohesion. Sections zip around like shoals of sardines. The conductor does not have to overwork as a result. He displays Karajan-like composure, tugging and tweaking with rooted feet and fluid torso. Shostakovich often requires juice, but Jansons' basic stance does not shift. Rather, a discernible grimace flashes over his face, and his taut, lupine frame bristles with concentrated commitment.

The playing lost some of its granite-like resolve in the second and third movements' elaborate yarn. It was difficult to detect the sense of direction evident before, but the Moderato nevertheless sneered and the Adagio was desolate. These passages were above all a potpourri of glittering technique. Peering into a musical workshop, we observed the way Jansons would manage the joins between sections with elasticity, or would flash a new image into view like switching stills. The main thrust was always horizontal, but the ear would also wallow in a wealth of detail, from a chirruping gauze of winds to the perfectly balanced mating song of antiphonal violins.

The raw materials are exceptional: earthy, coursing strings, piquant winds and a brass section of molten intensity. The venom with which the timpanist attacks his instrument makes you wonder how long the skins will withstand the assault.

Whether this work speaks of the terrors of Naziism, as the original programme had it, or the perils of Stalinism, as Shostakovich's memoirs hint, the seemingly triumphant ending is laced with ambiguity. Jansons delivered an apocalyptic vision, clicking the tempo into double time to evoke a juggernaut ploughing into view.

The secrets to this orchestra's success are clearly various. Chief among them is an all-round embodiment of the house style. There is commitment in playing and chemistry within the whole. Those that pour over their fingerboards with the greatest ardour tend to be the older players, guarding a stylistic heirloom and passing it on. When the orchestra finally stood to receive indebted applause, its players beamed a brazen sense of purpose.