Magnetic double perovskite (DP) compounds provide a fertile playground to explore interesting electronic and magnetic properties. By complementary macroscopic characterizations, neutron powder diffraction measurements, and first-principles calculations, we have performed comprehensive studies on the magnetic ordering in the DP compound Sr₂NiWO₆. It is found by neutron diffraction to order magnetically in a collinear type-II antiferromagnetic (AFM) structure in a tetragonal lattice with k = (0.5, 0, 0.5) below T_N = 56 K. In the ground state, the ordered moment of the spin-1 Ni²⁺ ions is determined to be 1.9(2) μ_B, indicating a significant quenching of the orbital moment. The Ni²⁺ moments in Sr₂NiWO₆ are revealed to cant off the c axis by 29.2°, which is well supported by the first-principles magnetic anisotropy energy calculations. Furthermore, the in-plane and out-of-plane next-nearest-neighbor superexchange couplings (J₂ and J_2c) are found to play a dominant role in the spin Hamiltonian of Sr₂NiWO₆, which accounts for the stabilization of the type-II AFM structure as its magnetic ground state.