I will discuss a new product structure on the two-sided bar constructions of singular cochains. This bar construction is used in the Eilenberg-Moore theorem to compute the cohomology of pull-backs of fibrations. The product structure is based on so-called homotopy Gerstenhaber operations on singular cochains, transforming the two-sided bar construction into an $A_\infty$-algebra. This type of algebra is associative up to a strong form of homotopy. The new product includes those previously defined by Baues, Gerstenhaber-Voronov, Kadeishvili-Saneblidze, and Carlson-Franz as special cases. Consequently, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is elevated to a quasi-isomorphism of $A_\infty$-algebras.