In L₂(R^d;Cⁿ), we consider a wide class of matrix elliptic operators A_ε of order 2p (where p≥2) with periodic rapidly oscillating coefficients (depending on x/ε). Here ε > 0 is a small parameter. We study the behavior of the operator exponent e^-Aετ for τ > 0 and small ε. We show that the operator e^-Aετ converges as ε → 0 in the operator norm in L₂(R^d;Cⁿ) to the exponent e^-A0τ of the effective operator A⁰. Also we obtain an approximation of the operator exponent e^-Aετ in the norm of operators acting from L₂(R^d;Cⁿ) to the Sobolev space H^p(R^d; Cⁿ). We derive estimates of errors of these approximations depending on two parameters: ε and τ. For a fixed τ > 0 the errors have the exact order O(ε). We use the results to study the behavior of a solution of the Cauchy problem for the parabolic equation ∂_τu_ε(x,τ)= -(A_ε u_ε)(x,τ)+F(x,τ) in R^d.