We report the first observation of rapidity-odd directed flow ($v_{1}$) for $D^{0}$ and $\overline{D^{0}}$ mesons in Au+Au collisions at $\sqrt{s_{NN}}$ = 200~GeV using the STAR detector at the Relativistic Heavy Ion Collider. The $v_{1}$-slope ($dv_{1}/dy$) of $D^{0}$ mesons is found to be significantly larger than that of light quark hadrons with the same sign measured at the same collision energy. The observed large $dv_{1}/dy$ of $D^{0}$ mesons implies its sensitivity to the initially tilted matter distribution. The data can be qualitatively explained by hydrodynamic model calculations.