We consider central simple $K$-algebras which happen to be differential graded $K$-algebras. Two such algebras $A$ and $B$ are considered equivalent if there are bounded complexes of finite dimensional $K$-vector spaces $C_A$ and $C_B$ such that the differential graded algebras $A\otimes_K \End_K^\bullet(C_A)$ and $B\otimes_K \End_K^\bullet(C_B)$ are isomorphic. Equivalence classes form an abelian group, which we call the dg Brauer group. We prove that this group is isomorphic to the ordinary Brauer group of the field $K$.