The layered crystal of EuSn$_2$As$_2$ has a Bi$_2$Te$_3$-type structure in rhombohedral ($Rbar{3}m$) symmetry and has been confirmed to be an intrinsic magnetic topological insulator at ambient conditions. Combining {it ab initio} calculations and emph{in-situ} x-ray diffraction measurements, we identify a new monoclinic EuSn$_2$As$_2$ structure in $C2/m$ symmetry above $sim$14 GPa. It has a three-dimensional network made up of honeycomb-like Sn sheets and zigzag As chains, transformed from the layered EuSn$_2$As$_2$ via a two-stage reconstruction mechanism with the connecting of Sn-Sn and As-As atoms successively between the buckled SnAs layers. Its dynamic structural stability has been verified by phonon mode analysis. Electrical resistance measurements reveal an insulator-metal-superconductor transition at low temperature around 5 and 15 GPa, respectively, according to the structural conversion, and the superconductivity with a textit{T}${rm {_C}}$ value of $sim 4$ K is observed up to 30.8 GPa. These results establish a high-pressure EuSn$_2$As$_2$ phase with intriguing structural and electronic properties and expand our understandings about the layered magnetic topological insulators.
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