Parabolic-type problems, involving a variational complementarity formulation, arise in mathematical models of several applications in Engineering, Economy, Biology and different branches of Physics. These kinds of problems present several analytical and numerical difficulties related, for example, to time evolution and a moving boundary. We present a numerical method that employs a global convergent nonlinear complementarity algorithm for solving a discretized problem at each time step. Space discretization was implemented using both the finite difference implicit scheme and the finite element method. This method is robust and efficient. Although the present method is general, at this stage we only apply it to two one-dimensional examples. One of them involves a parabolic partial differential equation that describes oxygen diffusion problem inside one cell. The second one corresponds to a system of nonlinear differential equations describing an in situ combustion model. Both models are rewritten in the quasi-variational form involving moving boundaries. The numerical results show good agreement when compared to direct numerical simulations.