When the Cassini spacecraft flew by Titan on 13 June 2007, at 13.6 Saturn local time, Titan was directly observed to be outside Saturn′s magnetopause. Cassini observations showed dramatic changes of magnetic field orientation as well as other plasma flow parameters during the inbound and outbound segments. In this paper, we study Titan's ionospheric responses to such a sudden change in the upstream plasma conditions using a sophisticated multispecies global MHD model. Simulation results of three different cases (steady state, simple current sheet crossing, and magnetopause crossing) are presented and compared against Cassini Magnetometer, Langmuir Probe, and Cassini Plasma Spectrometer observations. The simulation results provide clear evidence for the existence of a fossil field that was induced in the ionosphere. The main interaction features, as observed by the Cassini spacecraft, are well reproduced by the time-dependent simulation cases. Simulation also reveals how the fossil field was trapped during the interaction and shows the coexistence of two pileup regions with opposite magnetic orientation, as well as the formation of a pair of new Alfven wings and tail disconnection during the magnetopause crossing process.