When Simon Woods, CEO of the Los Angeles Philharmonic, came on stage to announce that Gustavo Dudamel would not be conducting the concert due to a "family emergency", there was a slight sense of disappointment in the Walt Disney Concert Hall audience. But when Woods introduced and extolled the promise of Dudamel's last-minute replacement, the crowd was already in Stephen Mulligan's corner when he came onto stage himself. From then on it was like they were rooting not only for the two young soloists – and Beethoven, of course – but for the young LA Phil Conducting Fellow, as if he were a hometown hero. The orchestra played for Mulligan as if he were a maestro.

The winds were super, the strings (with divided violins) were more sparing of their bow length than usual, and the balances overall, full of color and shot through with light, were qualities you might expect from period instrument orchestras. The French horns were glorious, and from his commanding position, high up stage left, the timpanist with small hard-headed drumsticks occasionally took over the action like he was beating for the galley slaves in Ben Hur, including melodramatically articulated, introductory flourishes to the cadenzas.

Conrad Tao in the First Concerto was knowing, bright- eyed and bushy-tailed, choosing to go with a modified, often ingenious and winning, fleet, exhilarating Early Classical style. Whether he was playing with legato sweep or a range of staccato and separé notes, he always made sure that the notes were given full value, and that he was in synch with the orchestra. Tao was similarly resourceful in dealing with the biggest of the three Beethoven cadenzas for the first movement, throwing in some marvelously deconstructed bars and phrases here and there, a toy piano sound for the march just when it was needed, and just before the end, a few extra seconds of glittering trills. As brilliant and knowing as it was, however, Tao's way with Beethoven's cadenza began to seem long, and eventually disconnected from the music that had started it all. Beethoven would have been better served if Tao had just gone ahead and written his own.

For his encore, Tao returned without his jacket to rip into Elliott Carter's Caténaires, a four-minute burst of manic energy from 2006 which may have been intended, as the composer said, "to produce a wide variety of expression," but which was mostly just totally mad.

Although Mulligan started off in the Third Concerto with the same brisk speed they had employed for the First, it was obvious at once that this was a different world. The orchestra sounded lusher, even the timpani seemed gentler. And when Beatrice Rana made her first dramatic entrance, bubbling up before the orchestra had finished its introduction, it turned out to be less an intervention than a joining up of forces, Rana starting almost at once to work with the conductor and orchestra to expand and relax the tempo to give her room in which to do more than just churn out notes. She began sustaining multi-bar phrases with a singing, nuanced, varied tone which in turn inspired the orchestra including some wonderful flute playing. She structured long runs in a natural and organic way to shape and energize the music's forward motion. But it wasn't until the massive first movement cadenza that she let us glimpse her true power, her ability to sculpt and move massive amounts of sound – and music. The second half of the cadenza was pure wizardry, so hypnotically did Rana hold us in her spell, until she and the orchestra drew the movement to a close with an unusually in-touch, post-cadenza dialogue.

Further proof that Rana and the orchestra were now on the same wavelength came with her slow, hushed playing of the Largo's opening theme, her simple elegance, her magical dialogue with flute and bassoon, the way she tore into the Rondo, and the way the cellos nailed the little fugue in F minor – tight, precise, dynamic. When the audience demanded an encore, Rana played Chopin Étude in C minor with wild abandon, Op.25 no. 12.

The concert had opened with a thoughtful Egmont Overture which gave everyone a chance to get acquainted before the two concertos took over.