  Let $A$ be a monomial associative finite dimensional algebra over a field
$\Bbbk$ of characteristic zero. It is well known that the Hochschild cohomology
of $A$ can be computed using Bardzell's complex $B(A)$. The aim of this article
is to describe an explict $L_\infty$-structure on $B(A)$ that induces a weak
equivalence of $L_\infty$-algebras between $B(A)$ and the Hochschild complex
$C(A)$ of $A$. This allows us to describe the Maurer-Cartan equation in terms
of elements of degree $2$ in $B(A)$. Finally, we make concrete computations
when $A$ is a truncated algebra, and we prove that Bardzell's complex for
radical square zero algebras is in fact a dg-Lie algebra.
