Gauge PDEs are geometrical objects underlying local gauge field theories. If one disregards Lagrangians, gauge PDEs give a rather flexible and geometrical way to encode all the structures of the system including a specific version of its Batalin-Vilkovisky (BV) formulation. Other way around, conventional BV formulation can be identified as a very special gauge PDE equipped with extra structures. A remarkable feature of this approach is that there is no need to define a system in terms of jet-bundles (off-shell terms). Instead, one can start with an equation manifold or its extensions or gauge quotients, reproducing the familiar geometrical approach to PDEs in the case on non-gauge systems. However, just like in the case of usual PDEs it is difficult to encode the BV extension of the Lagrangian in terms of the intrinsic geometry of the equation manifold while working on jet-bundles is often very restrictive (e.g. in analyzing boundary behavior in the context of AdS/CFT correspondence). We show that BV Lagrangian (or its weaker analogs) can be encoded in the compatible graded presymplectic structure on the gauge PDE. In the case of genuine Lagrangian systems this presymplectic structure is related to a certain completion of the canonical BV symplectic structure. A presymplectic gauge PDE gives rise to the BV gauge system through an appropriate generalization of the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) sigma-model construction followed by taking the symplectic quotient and resulting in the usual 1st order BV formulation or its presymplectic extension. The approach is illustrated on the standard examples of gauge theories with an emphasis on the (conformal) gravity.