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By applying a hybrid (kinetic ions and fluid electrons) simulation code, we study the plasma environment of Saturn's largest moon Titan during Cassini's T96 flyby on 1 December 2013. The T96 encounter marks the only observed event of the entire Cassini mission where Titan was located in the supersonic solar wind in front of Saturn's bow shock. Our simulations can quantitatively reproduce the key features of Cassini magnetic field and electron density observations during this encounter. We demonstrate that the large-scale features of Titan's induced magnetosphere during T96 can be described in terms of a steady state interaction with a high-pressure solar wind flow. About 40min before the encounter, Cassini observed a rotation of the incident solar wind magnetic field by almost 90 degrees. We provide strong evidence that this rotation left a bundle of fossilized magnetic field lines in Titan's ionosphere that was subsequently detected by the spacecraft.