The problem of energy kinetics of the harmonic oscillator under the influence of the colored noise is studied in a novel approach that describes the evolution by a discrete time random walk with randomly varying step. In this approach the variations of the oscillator's energy on adjacent time intervals happen to be virtually uncorrelated even for large correlation times of the noise. The average time of the first passage of the oscillator with the initial zero energy across some threshold value is calculated. The pre-exponent factor of transition rate is found to depend on the parameters of noise and not on oscillator damping and correctly describes the case of zero friction. The agreement in exponential factors obtained by the suggested approach and kinetic equation is demonstrated for narrow-band colored noise.