Many mathematical models (gas dynamics, population dynamics and others) use partial differential equations with nonlinearity in the differentiation operator. In addition to nonlinearity, the model can be complicated by fractional derivatives, delay effects, the presence of several spatial variables. In this paper, we focus on the nonlinearity in the heat conductivity coefficient and delay effects. A one-dimensional quasilinear parabolic equations with delay effects are considered. Equations of this kind arise, for example, during high-temperature processes in plasma, where the heat conductivity coefficient is a nonlinear function of temperature. In view of the special form of nonlinearity (quasilinearity), it is possible to construct an effective algorithm for solving the equations under consideration. It consists in constructing a linear (relative to the values of functions at the next time layer) implicit difference scheme. The values of the heat conductivity coefficient are calculated on the previous time layer, thus we obtain a tridiagonal system. The obtaining system of linear equations has a tridiagonal dominance and is solved by the sweep method. The delay effects are taken into account using the interpolation and extrapolation of discrete prehistory. The order of approximation error for the constructed method and order of convergence are studied. The convergence order theorem is obtained, which uses the methods from the general theory of difference schemes and discrete analog of Gronwall’s lemma.