Yes, absolutely. I really think the best place for learning this is the book by Loday and Vallette.
This example fits more generally into the following context. Let $P$ be a Koszul operad and $A$ a $P$-algebra. There is attached to $P$ a cohomology theory for $P$-algebras. The two most well known examples is that when $P = \mathsf{Lie}$ we get ordinary Lie algebra cohomology, and that when $P = \mathsf{Ass}$ we get Hochschild homology. This is in general defined by the chain complex $$ \mathrm{Hom}_{\mathbb S}(P^!,\mathrm{End}(A)) $$ i.e. maps of $\mathbb S$-modules from the Koszul dual co-operad of $P$ to the endomorphism operad of $A$. This chain complex sits inside $$ \mathrm{Hom}_{\mathbb k}(P^!,\mathrm{End}(A)), $$ i.e. maps which are not necessarily equivariant. The latter space is itself in a natural way the sum of all components of an operad, the convolution operad. Convolution gives an operad structure on maps from any co-operad to an operad.
Now on the sum of all components of an operad there is a pre-Lie structure given by operadic composition. Antisymmetrizing this gives an honest Lie bracket. This bracket on $$ \mathrm{Hom}_{\mathbb k}(P^!,\mathrm{End}(A)), $$ leaves $$ \mathrm{Hom}_{\mathbb S}(P^!,\mathrm{End}(A)) $$ invariant, giving a Lie bracket on the chain complex for the operadic cohomology. This is what you are seeing.