Exact analysis of spin crossover in infinite elastic chain of two-spin exchange clusters was performed theoretically beyond mean-field approximation. Statistical mechanics of the system was exactly calculated by means of the transfer matrix technique. A possibility of spin-Peierls-like magneto-structural transition in a one-dimensional chain was shown as a result of interrelation of spin and elastic subsystems of the chain. The importance of the boundary conditions for spin crossover to occur was shown. The smoothness of spin crossover depends on the parameters of chain elasticity and the crossover position on the temperature scale is defined by the dependence of exchange integral on the cluster deformation. The proposed model qualitatively describes the main scenario of spin crossover effect in the chain polymer heterospin complex of copper(ii) hexafluoroacetylacetonate with methyl pyrazol-substituted nitronyl nitroxide containing two-spin exchange clusters.