Summary: Given an associative graded algebra equipped with a degree \(+1\) differential \(\Delta\) we define an \(A_\infty\)-structure that measures the failure of \(\Delta\) to be a derivation. This can be seen as a non-commutative analog of generalized BV-algebras. In that spirit we introduce a notion of associative order for the operator \(\Delta\) and prove that it satisfies properties similar to the commutative case. In particular when it has associative order 2 the new product is a strictly associative product of degree \(+1\) and there is compatibility between the products, similar to ordinary BV-algebras. We consider several examples of structures obtained in this way. In particular we obtain an \(A_\infty\)-structure on the bar complex of an \(A_\infty\)-algebra that is strictly associative if the original algebra is strictly associative. We also introduce strictly associative degree \(+1\) products for any degree \(+1\) action on a graded algebra. Moreover, an \(A_\infty\)-structure is constructed on the Hochschild cocomplex of an associative algebra with a non-degenerate inner product by using Connes’ B-operator.