  We apply the modern Batalin-Vilkovisky quantization techniques of Costello
and Gwilliam to noncommutative field theories in the finite-dimensional case of
fuzzy spaces. We further develop a generalization of this framework to theories
that are equivariant under a triangular Hopf algebra symmetry, which in
particular leads to quantizations of finite-dimensional analogs of the field
theories proposed recently through the notion of `braided $L_\infty$-algebras'.
The techniques are illustrated by computing perturbative correlation functions
for scalar and Chern-Simons theories on the fuzzy $2$-sphere, as well as for
braided scalar field theories on the fuzzy $2$-torus.
