A complete first-principles bottom-up computational study of the magnetic properties of [Cu(pz)2](ClO4)2 is presented. A remarkable agreement is observed in the whole range of temperatures between simulated and experimental magnetic susceptibility data. Interestingly, the simulated heat capacity values show an anomaly close to the Néel temperature of 4.21 K associated with a transition from a two-dimensional (2D) antiferromagnet to a three-dimensional (3D) ordered state. The antiferromagnetic behavior of [Cu(pz)2](ClO4)2 is due to a 2D magnetic topology owing to two antiferromagnetic JAB interactions through pyrazine ligands. Although presenting a very similar molecular arrangement, the numerical values of the two magnetically significant JAB couplings differ by 25% (-10.2 vs -7.3 cm(-1)). This difference can be ascribed to three main contributions: (i) the central pyrazine ring shearing-like distortion, (ii) the effect of the orientation of the perchlorate counterions, and (iii) a hitherto unrecognized skeleton-counterion cooperation arising from different hydrogen bonding contributions in the two most significant JAB couplings. The impact of the orientation of the perchlorate counterions is disclosed by comparison to JAB studies using structurally similar ligands but with different electronegativity (namely, BF4(-), BCl4(-), and BBr4(-)). Pyrazine ligands and perchlorate counterions prove to be noninnocent.