We give a criterion for the complete reducibility of modules satisfying a composability condition for a meromorphic open-string vertex algebraVVusing the first cohomology of the algebra. For aVV-bimoduleMM, letH^∞1(V,M)\hat {H}^{1}_{\infty }(V, M)be the first cohomology ofVVwith the coefficients inMM, which is canonically isomorphic to the quotient space of the space of derivations fromVVtoWWby the subspace of inner derivations. LetZ^∞1(V,M)\hat {Z}^{1}_{\infty }(V, M)be the subspace ofH^∞1(V,M)\hat {H}^{1}_{\infty }(V, M)canonically isomorphic to the space of derivations obtained from the zero mode of the right vertex operators of weight11elements such that the difference between the skew-symmetric opposite action of the left action and the right action on these elements are Laurent polynomials in the variable. IfH^∞1(V,M)=Z^∞1(V,M)\hat {H}^{1}_{\infty }(V, M)= \hat {Z}^{1}_{\infty }(V, M)for everyZ\mathbb {Z}-gradedVV-bimoduleMM, then every leftVV-module satisfying a composability condition is completely reducible. In particular, since a lower-boundedZ\mathbb {Z}-graded vertex algebraVVis a special meromorphic open-string vertex algebra and leftVV-modules are in fact what has been called generalizedVV-modules with lower-bounded weights (or lower-bounded generalizedVV-modules), this result provides a cohomological criterion for the complete reducibility of lower-bounded generalized modules for such a vertex algebra. We conjecture that the converse of the main theorem above is also true. We also prove that when a grading-restricted vertex algebraVVcontains a subalgebra satisfying some familiar conditions, the composability condition for grading-restricted generalizedVV-modules always holds and we needH^∞1(V,M)=Z^∞1(V,M)\hat {H}^{1}_{\infty }(V, M)= \hat {Z}^{1}_{\infty }(V, M)only for everyZ\mathbb {Z}-gradedVV-bimoduleMMgenerated by a grading-restricted subspace in our complete reducibility theorem.