Summary: We prove a Cohen-Macaulay version of a result by L. L. Avramov and E. S. Golod [Math. Notes 9, 30–32 (1971; Zbl 0222.13014)] and A. Frankild and P. Jørgensen [Isr. J. Math. 135, 327–353 (2003; Zbl 1067.13013)] about Gorenstein rings, showing that if a noetherian ring \(A\) is Cohen-Macaulay, and \(a_1, \ldots, a_n\) is any sequence of elements in \(A\), then the Koszul complex \(K(A; a_1, \ldots, a_n)\) is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring \(A\), by finding a Cohen-Macaulay DG-ring \(B\) such that \(\operatorname{H}^0(B) = A\), and using the Cohen-Macaulay structure of \(B\) to deduce results about \(A\). As application, we prove that if \(f : X \to Y\) is a morphism of schemes, where \(X\) is Cohen-Macaulay and \(Y\) is nonsingular, then the homotopy fiber of \(f\) at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given.