  We synthesized single crystals of composition Ba$_2$CuSi$_2$O$_6$Cl$_2$ and
investigated its quantum magnetic properties. The crystal structure is closely
related to that of the quasi-two-dimensional (2D) dimerized magnet
BaCuSi$_2$O$_6$ also known as Han purple. Ba$_2$CuSi$_2$O$_6$Cl$_2$ has a
singlet ground state with an excitation gap of ${\Delta}/k_{\rm B}\,{=}\,20.8$
K. The magnetization curves for two different field directions almost perfectly
coincide when normalized by the $g$-factor except for a small jump anomaly for
a magnetic field perpendicular to the $c$ axis. The magnetization curve with a
nonlinear slope above the critical field is in excellent agreement with
exact-diagonalization calculations based on a 2D coupled spin-dimer model.
Individual exchange constants are also evaluated using density functional
theory (DFT). The DFT results demonstrate a 2D exchange network and weak
frustration between interdimer exchange interactions, supported by weak
spin-lattice coupling implied from our magnetostriction data. The
magnetic-field-induced spin ordering in Ba$_2$CuSi$_2$O$_6$Cl$_2$ is described
as the quasi-2D Bose-Einstein condensation of triplets.
