We present broad-band observations and analysis of Swift gamma-ray burst (GRB) 120119A. Our early-time afterglow detections began under 15 s after the burst in the host frame (redshift z = 1.73), and they yield constraints on the burst energetics and local environment. Late-time afterglow observations of the burst show evidence for a moderate column of dust (A$_V$ ≈ 1.1 mag) similar to, but statistically distinct from, dust seen along Small Magellanic Cloud sightlines. Deep late-time observations reveal a dusty, rapidly star-forming host galaxy. Most notably, our early-time observations exhibit a significant red-to-blue colour change in the first ̃200 s after the trigger at levels heretofore unseen in GRB afterglows. This colour change, which is coincident with the final phases of the prompt emission, is a hallmark prediction of the photodestruction of dust in GRB afterglows. We test whether dust-destruction signatures are significantly distinct from other sources of colour change, namely a change in the intrinsic spectral index β. We find that a time-varying power-law spectrum alone cannot adequately describe the observed colour change, and allowing for dust destruction (via a time-varying A$_V$) significantly improves the fit. While not definitively ruling out other possibilities, this event provides the best support yet for the direct detection of dust destruction in the local environment of a GRB.