Abstract: To reconcile the current tensions between high and low redshift observations, we perform the first constraints on the Finslerian cosmological models including the effective dark matter and dark energy components. We find that all the four Finslerian models could alleviate effectively the Hubble constant (H-0) tension and the amplitude of the root-mean-square density fluctu-ations (s(8)) tension between the Planck measurements and the local Universe observations at the 68% confidence level. The addition of a massless sterile neutrino and a varying total mass of active neutrinos to the base Finslerian two-parameter model, respectively, reduces the H-0 tension from 3.4s to 1.9s and alleviates the s8 tension better than the other three Finslerian models. Computing the Bayesian evidence, with respect to ACDM model, our analysis shows a weak preference for the base Finslerian model and moderate preferences for its three one-parameter extensions. Based on the model-independent Gaussian Processes, we propose a new linear relation which can describe the current redshift space distortions data very well. Using the most stringent constraints we can provide, we have also obtained the limits of typical model parameters for three one-parameter extensional models.