As a new antiperovskite nitride, ZnFe3N was synthesized and characterized by almost completely substituting iron atoms at corner positions of γ'-Fe4N. The magnetic interactions of the system with the space group Pm3[combining macron]m are fully investigated. The critical behavior was investigated based on the measured magnetic data around the ferromagnetic phase transition temperature. In this work, the values of critical exponents (β, γ and δ) were obtained systematically using the Kouvel-Fisher method in the critical region. The Widom scaling law (δ = 1 + γβ-1) and the scaling equation (m = f±(h)) were used to reveal the reliability of these values. The values of the critical exponents (β = 0.325, γ = 1.228, and δ = 4.778) are different from those predicted by the three-dimensional (3D) Heisenberg model and mean-field model, and are very close to those of the 3D-Ising model. Combined with ESR analysis, the spin clusters induced by changes in chemical bonds are considered to be the cause for the existence of an anisotropic short-range ordered state in this ferromagnetic system.