Jones claimed earlier in the week that Scotland were "red-hot" favourites to beat England who, missing the likes of Owen Farrell and Courtney Lawes through injury, fielded seven players boasting 10 or fewer caps.
Nevertheless, the visitors controlled possession and territory throughout the first half, with Smith breaking the deadlock in the 17th minute.
The hosts responded almost immediately with Darcy Graham embarking on a powerful surge, before offloading for White – on while Ali Price received treatment for a head injury – to cross just minutes into his Test career.
The right boot of Smith kept England within touching distance at 10-9 behind, and he gratefully received Ben Youngs' offload 13 minutes into the second half; powering to the line after a solid spell of pressure.
But momentum swung in favour of Scotland, who were awarded a penalty try after the TMO ruled Cowan-Dickie deliberately knocked Russell's punt into touch with Graham lurking.
With Cowan-Dickie subsequently sin-binned, the hosts capitalised on their numerical advantage with the boot of Russell edging them back in front, and they held on despite a late surge from their opponents.
