Gregor Townsend's side overturned a 15-point deficit, having trailed 20-5 and later 23-12, before rallying superbly after the break. Replacement hooker George Turner sealed the win with a decisive try in the 74th minute, completing a remarkable turnaround.
Although Scotland had crossed earlier through Kyle Steyn, Darcy Graham and Finn Russell, it was Turner's late score that finally put the visitors in front for the first time in the match. Russell, who finished with 11 points, played a key role throughout, with his vision and creativity helping swing the contest Scotland's way.
Graham's try proved the pivotal moment, sparked by a moment of brilliance from Russell and a lapse in the Welsh defence following a restart.
Wales had looked in control after tries from Rhys Carre and Josh Adams, along with 10 points from Sam Costelow, but Scotland's resilience and attacking intent ultimately told.
The win keeps Scotland firmly in the hunt for silverware. Victory over Ireland in Dublin on 14 March would secure a first Triple Crown since 1990.