  We show that, up to Morita equivalence, any finite-dimensional algebra with a
suitable homological system, admits an exact Borel subalgebra. This generalizes
a theorem by Koenig, K\"ulshammer and Ovsienko, which holds for
quasi-hereditary algebras. Our proof follows the same general scheme proposed
by these authors, in a more general context: we associate a differential graded
tensor algebra with relations, using the structure of $A_{\infty}-$algebra of a
suitable Yoneda algebra, and use its category of modules to describe the
category of filtered modules associated to the given homological system.
