We study Abelian Maxwell-Chern-Simons theory in three-dimensional AdS black hole backgrounds for both integer and non-integer Chern-Simons coupling. Such theories can be derived from various string theory constructions, which we review in the present work. In particular we find exact solutions in the low frequency, low momentum limit, ω, k ≪ T (hydrodynamic limit). Using the holographic principle, we translate our results into correlation functions of vector and scalar operators in the dual strongly coupled 1+1-dimensional quantum field theory with a chiral anomaly at non-zero temperature T. Starting from the conformal case we show applicability of the hydrodynamic limit and discuss extensions to the non-conformal case. Correlation functions in the conformal case are compared to an exact field theoretic computation.