Classical and quantum phase transitions (QPTs), with their accompanying concepts of criticality and universality, are a cornerstone of statistical thermodynamics. An excellent example of a controlled QPT is the field-induced ordering of a gapped quantum magnet. Although numerous "quasi-one-dimensional"coupled spin-chain and -ladder materials are known whose ordering transition is three-dimensional (3D), quasi-two-dimensional (2D) systems are special for multiple reasons. Motivated by the ancient pigment Han purple (BaCuSi₂O₆), a quasi-2D material displaying anomalous critical properties, we present a complete analysis of Ba_0.9Sr_0.1CuSi₂O₆. We measure the zero-field magnetic excitations by neutron spectroscopy and deduce the spin Hamiltonian. We probe the field-induced transition by combining magnetization, specific-heat, torque, and magnetocalorimetric measurements with nuclear magnetic resonance studies near the QPT. With a Bayesian statistical analysis and large-scale Quantum Monte Carlo simulations, we demonstrate unambiguously that observable 3D quantum critical scaling is restored by the structural simplification arising from light Sr substitution in Han purple.