A mixed-classical-quantum approach to describe laser-induced excitation, fragmentation and, in particular, ionization of one-electron molecular dimers is presented. Self-consistent equations of motion for the classical nuclear dynamics and the quantum-mechanical electronic evolution are derived. They are solved using a combined set of basis functions for the electronic wave function consisting of atomic orbitals and chainlike arranged Gaussians to account for a realistic description of the continuum dynamics of the ionized electron. For the molecular benchmark system H-2(+), the obtained results are in good agreement with those of full-quantum-mechanical calculations, which are available for relatively short laser pulses. For longer pulses, the present calculation signals a transient effect of molecular stabilization against fragmentation at corresponding (low) laser intensities.