In order to investigate the electron dynamics at the alizarin/I-2(-)/TiO2 interface this study uses a novel state-of-the-art quantum-classical approach that combines time-dependent density functional theory with surface hopping in the Kohn-Sham basis. Representing the dye-sensitized semiconductor Gratzel cell with the I-/I-3(-) mediator, the system addresses the problems of an organic/inorganic, molecule/bulk interface that are commonly encountered in molecular electronics, photovoltaics, and photoelectrochemistry. The processes studied include the relaxation of the injected electron inside the TiO2 conduction band (CB), the back electron transfer (ET) from TiO2 to alizarin, the ET from the surface to the electrolyte, and the regeneration of the neutral chromophore by ET from the electrolyte to alizarin. Developing a theoretical understanding of these processes is crucial for improving solar cell design and optimizing photovoltaic current and voltage. The simulations carried out for the entire system that contains many electronic states reproduce the experimental time scales and provide detailed insights into the ET dynamics. In particular, they demonstrate the differences between the optimized geometric and electronic structure of the system at 0 K and the experimentally relevant structure at ambient temperature. The relaxation of the injected electron inside the TiO2 CB, which affects the solar cell voltage, is shown to occur on a 100 fs time scale and occurs simultaneously with the electron delocalization into the semiconductor bulk. The transfer of the electron trapped at the surface to the ground state of alizarin proceeds on a 1 ps time scale and is facilitated by vibrational modes localized on alizarin. If the electrolyte mediator is capable of approaching the semiconductor surface, it can form a stable complex and short-circuit the cell by accepting the photoexcited electron on a subpicosecond time scale. The ET from TiO2 to both alizarin and the electrolyte diminishes the solar cell current. Finally, the simulations show that the electrolyte can efficiently regenerate the neutral chromophore. This is true even though the two species do not form a chemical bond and, therefore, the electronic coupling between them is weaker than in the TiO2-chromophore and TiO2-electrolyte donor-acceptor pairs. The chromophore-electrolyte coupling can occur both directly through space and indirectly through bonding to the semiconductor surface. The ET events involving the electrolyte are promoted primarily by the electrolyte vibrational modes.