In this talk, by considering gauge theories induced by finite spectral triples, we aim to explain how the BV formalism naturally inserts in the framework of noncommutative geometry. Reaching this goal entails that not only all the steps of the BV construction, from the introduction of ghost/anti-ghost fields to the construction of the gauge-fixed BRST complex, can be expressed using noncommutative geometric objects, but that the entire construction has an intrinsically noncommutative geometric nature. Moreover, for the first time we relate the BRST complex to another cohomological theory, naturally appearing in noncommutative geometry: the Hochschild complex of a graded algebra.