A method for the numerical solution of the dissipative Liouville-von Neumann equation is presented. The reduced density operator is expanded in a basis of time-dependent wave functions. This guarantees that the size of the basis required in the representation of the density operator is minimal. Equations of motion for the expansion wave functions and the density matrix elements are obtained from the Dirac-Frenkel variational principle. The numerical effort of the method scales proportional to N, where N is the dimension of the relevant Hilbert space. As a first example, the dynamics of a three-mode system with vibronic coupling (with N approximate to 50 000) coupled to a dissipative bath has been studied, modeling the S-1/S-2 states of pyrazine. For the cw-absorption spectrum, fast convergence with respect to the number of expansion wave functions has been obtained.