The coloured noise induced escape rate from the lower energy stable state of a driven nonlinear microcavity oscillator has been investigated by means of quasi-classical kinetic equations. We show that for coloured, i.e. narrow-band, relatively intense noise, the escape time is controlled by the interplay of two mechanisms: the noise induced drift and adiabatic regular shift of the oscillator state towards unstable saddle point. The cross-over between these mechanisms takes place in a particular range of the driving field intensity values, depending on the ratio between the oscillator damping and the coloured noise spectrum width. The dependence of the transition rate on the noise correlation time is analyzed for wide range of correlation time values.