  The layered crystal of EuSn$_2$As$_2$ has a Bi$_2$Te$_3$-type structure in
rhombohedral ($R\bar{3}m$) 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.
