A variational formulation of the problem of unsteady-state heat conduction is presented. A non-linear functional obtained as a result of a thermodynamic analysis of the processes of heat transfer in unsteady-state systems is suggested. The variational calculational method can be used for solving problems with a strong non-linearity for which finite-difference schemes do not allow one to obtain satisfactory results. To simplify the presentation, a non-linear one-dimensional problem is considered as an example. The functional can be generalized to the case of three-dimensional problems; as well as transformed for other coordinate systems. A technique for calculating the approximation error in variational calculus is suggested that makes it possible to determine approximations to the solution from below and from above.