Using it ab initio calculations, we investigate the magnetic ground states of quasi-one-dimensional insulating CrSbX₃ (X = S, Se) with infinite double-rutile chains. Within conventional band theory, without explicit Coulomb correlations (U), we obtain band gaps in close agreement with experiment. Remarkably, we find that the magnetic order is highly sensitive to the Cr-Cr bond length d_(rm Cr-Cr): increasing the bond length induces a transition from antiferromagnetic to ferromagnetic order at a critical distance dᶜ_(rm Cr-Cr) ≈ 3.53 (± 0.05) Å. Accordingly, CrSbS₃ lies near the transition boundary, whereas CrSbSe₃ is robustly ferromagnetic, in good agreement with experiment. Analysis of the exchange interactions reveals that the first-order phase transition is dominated by a sign reversal of the intrachain nearest-neighbor superexchange J₁ mediated by chalcogen ions, while the intrachain direct exchange J₂ remains ferromagnetic and changes only gradually. This behavior reflects an emergent Bethe-Slater-like behavior driven by competing exchange pathways in a quasi-1D transition-metal system, where the competition between J₁ and J₂ dictates the magnetic ground state. Besides, the electronic structures of the ground states of each compound are investigated.