A microscopic magnetic model for the spin-1/2 Heisenberg chain compound CuSe₂O₅ is developed based on the results of a joint experimental and theoretical study. Magnetic susceptibility and specific heat data give evidence for quasi-one-dimensional (1D) magnetism with leading antiferromagnetic (AFM) couplings and an AFM ordering temperature of 17 K. For microscopic insight, full-potential density functional theory (DFT) calculations within the local density approximation (LDA) were performed. Using the resulting band structure, a consistent set of transfer integrals for an effective one-band tight-binding model was obtained. Electronic correlations were treated on a mean-field level starting from LDA (LSDA+U method) and on a model level (Hubbard model). With excellent agreement between experiment and theory, we find that only two couplings in CuSe₂O₅ are relevant: the nearest-neighbour intra-chain interaction of 165 K and a non-frustrated inter-chain (IC) coupling of 20 K. From a comparison with structurally related systems (Sr₂Cu(PO₄)₂, Bi₂CuO₄), general implications for a magnetic ordering in presence of IC frustration are made.