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.
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