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We show that in free surface flows, a uniform, streamwise current over small-amplitude wavy bottom topography generates cross-stream drift velocity. This drift mechanism, referred to as the current-bathymetry interaction induced drift (CBIID), is specifically understood in the context of a simplified nearshore environment consisting of a uniform alongshore current, onshore propagating surface waves, and monochromatic wavy bottom making an oblique angle with the shoreline. CBIID is found to originate from the steady, non-homogeneous solution of the governing system of equations. Similar to Stokes drift induced by surface waves, CBIID also generates a compensating Eulerian return flow to satisfy the no-flux lateral boundaries, e.g. the shoreline. CBIID increases with an increase in the particle's initial depth, bottom undulation's amplitude, and the strength of the alongshore current. Additionally, CBIID near the free (bottom) surface increases (decreases) with an increase in bottom undulation's wavelength. Maximum CBIID is obtained for long wavelength bottom topography that approximately makes $π/4$ angle with the shoreline. Unlike Stokes drift, particle excursions due to current-bathymetry interactions might not be small, hence analytical expressions based on the small-excursion approximation could be inaccurate. We provide an alternative $z$-bounded approximation, which leads to highly accurate expressions for drift velocity and time period of particles especially located near the free surface. Realistic parametric analysis reveals that in some nearshore environments, CBIID's contribution to the net Lagrangian drift can be as important as Stokes drift, implying that CBIID can have major implications in cross-shelf tracer transport.