The averaging problem for convection-diffusion non-stationary parabolic operator with rapidly oscillating coefficients is studied. Under the assumptio- n that the coefficients are periodic in spatial variables and random stationar- y in time and that they possess certain mixing properties, we show that in appropriate moving coordinates the measures generated by the solutions of original problems converge weakly to a solution of limit stochastic PDE. The homogenized problem is well-posed and defines the limit measure uniquely.