Summary: In the current paper, we study a Petrov-Galerkin method for a Cauchy problem for an operator-differential equation with a monotone operator in a separable Hilbert space. The existence and the uniqueness of a strong solution of the Cauchy problem are proved. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The minimal requirements to the operators of the equation were demanded, which guaranteed the convergence of the approximate solutions. There were no assumptions of the structure of the operators. Therefore, the method, specified in this paper, can be applied to a wide class of the parabolic equations as well as to the integral-differential equations. The initial boundary value problem for nonlinear parabolic equations of the fourth order on space variables was considered as the application.