One key feature of the multi-orbital superconducting Sr₂RuO₄ is the presence of nodal or near-nodal quasiparticle excitations revealed in a wide variety of experiments. Typically, a nodal gap structure in a two-dimensional model would be inconsistent with the chiral or helical p-wave interpretations. However, we demonstrate that true gap nodes may emerge along certain high-symmetry directions on the quasi-one-dimensional Fermi surfaces, if the multi-orbital chiral or helical p-wave pairings acquire peculiar forms wherein the d_xz and d_yz orbitals develop k_y- and k_x-like pairings, respectively. Spin-orbit coupling η induces a near-nodal gap of order (η²/W²)Δ₀, where Δ₀ is the gap amplitude and W roughly the bandwidth. Provided the aforementioned pairing is predominant, the near-nodal gap structure is robust upon the inclusion of other multi-orbital pairings that share the same symmetries. In light of the recent experimental progresses, our proposal suggests that two-dimensional p-wave pairings may still be viable candidate ground states for Sr₂RuO₄. A near-nodal helical p-wave order, for example, would also be consistent with the substantial drop in the NMR Knight shift under an in-plane magnetic field.