Playing all of Bach's Sonatas and Partitas for solo violin from memory must be something like memorising an entire Dickens novel, then reciting it word perfect – complete with dramatic phrasing. Across two concerts, that's what the Russian violinist Alina Ibragimova is here to do at the Proms (the Bach, not the Dickens of course). The first in her pair of late night Proms was a utterly immersive experience.

In the 121st Proms season, it's surprising that any significant Bach firsts have yet to occur. Here was one: the first complete performances of the Violin Sonata no. 1 in G minor and the Partita no. 1 in B minor, with the Sonata no. 2 in A minor completing the sandwich. Ibragimova's style was assertive and unfussy, while her technique and musicianship stood up to the scrutiny invited by the exposure of solo performance. Ibragimova is well known for not using vibrato when performing these works, a style believed to be true to Bach's time.

Aptly, though she plays music from many other eras, the violin Ibragimova plays dates from c.1775, only 55 years after the pieces on tonight's programme were composed. The clean, raw quality of her interpretation was arresting, especially in fast arpeggiaic passages or with rapid double stops. Slower, quieter moments lacked some warmth, but this was entirely congruous with Ibragimova's pared-back colouring of moods throughout, including the restrained, expansive opening to the fulsome first sonata's Adagio. From here she soon grew into the busy fugue and lilting Siciliano. Particularly impressive were the demanding sections of double and multiple-stopping, even the grandest gestures executed with efficiency and lightness of touch. The Presto was taken swiftly and relatively roughly, without excessive colour.

This made an interesting contrast with the the Double: Presto in the B minor Partita, which was no less swift and still dazzling, but which had greater shape and control. This second work provided a great array of internal contrasts, containing double the number of movements of the others.  A bright Double, elegant Corrente and fiery aforementioned Double: Presto, prepared us for the alternating styles of the latter movements. A marvellously fun – yet as ever, unflamboyant – Borea (or bourree) heralded the final Double. There was a partial audience exodus at this point, presumably to catch trains, but rather a shame.

The third item on the programme, the Sonata no. 2 in A minor, had a subtler, different character. Notable for its relatively busy ornamentation, this Sonata contains a distinctive Andante. The violin is always accompanying itself to some extent in Bach's solo works, even by implying notes that our minds full in, but here it became a literal pulse underpinning the main melodies. Ibragimova was technically secure and precise, and maintained the energy through both parts to hypercritical effect. A complex and energised final Allegro ended the evening at its consistently high standard; spellbinding Baroque without any unnecessary flamboyance.

This a was breathtaking but not breathless concert, a brilliantly accomplished individual interpretation of what are arguably the less emotional of Bach's solo partitas and sonatas. If you prefer vibrato or you like a more overtly emotional performance, this wasn't for you. And of course, an entire Prom of one composer's work (even Bach) may be too intense for people who look to the Proms for its traditionally varied programmes. But personally, I can't wait to hear more in the second concert.