In this paper, we consider a modification of the fictitious domain method with continuation in terms of leading coefficients for a non-stationary nonlinear model of a non-Newtonian fluid in a limited three-dimensional domain. This modification is given as an auxiliary problem with a small parameter and in an extended (fictitious) region. And the model of the non-Newtonian problem is considered as the original problem. The purpose of this work is to prove the existence of any solutions of the auxiliary problem and their convergence to the corresponding solutions of the original problem as the small parameter tends to zero.Here we prove the existence of a generalized solution of the auxiliary problem and its convergence to the corresponding solution of the original problem. At the same time, for a strong solution, an estimate of the rate of convergence of a higher order is obtained, in comparison with the "classical" version of the method of fictitious regions. The main research methods are the methods of a priori estimates and Galerkin.