Synchronization is a fundamental problem in natural and artificial coupled multi-component systems. We investigate to what extent small-world couplings (extending the original local relaxational dynamics through the random links) lead to the suppression of extreme fluctuations in such systems. We use the framework of non-equilibrium surface growth to study and characterize the degree of synchronization in the system. In the absence of the random links, the surface in the steady state is "rough" (strongly de-synchronized state) and the average and the extreme height fluctuations diverge in the same power-law fashion with the system size (number of nodes). With small-world links present. the average size of the fluctuations becomes finite (synchronized state) and the extreme heights diverge only logarithmically in the large system-size limit. This latter property ensures synchronization in a practical sense in coupled multi-component autonomous systems with short-tailed noise and effective relaxation through the links. The statistics of the extreme heights is governed by the Fisher-Tippett-Gumbel distribution. We illustrate our findings through an actual synchronization problem in parallel discrete-event simulations.