In L₂(R^d;Cⁿ), consider a selfadjoint matrix second order elliptic differential operator B_ε, 0 < ε ≤ 1. The principal part of the operator is given in a factorized form, the operator contains first and zero order terms. The operator B_ε is positive definite, its coefficients are periodic and depend on x/ε. The behavior in the small period limit is studied for the operator exponential. The approximation in the (L₂ → L₂)-operator norm with error estimate of order O(ε2) is obtained. The corrector is taken into account in this approximation. The results are applied to homogenization of the solutions for the Cauchy problem for parabolic systems.