An exact sequence $0\to A\to B\to C\to 0$ of torsion-free abelian groups is quasi-balanced if the induced sequence $$0\to 𝐐\otimes Hom(X,A)\to 𝐐\otimes Hom(X,B)\to 𝐐\otimes Hom(X,C)\to 0$$ is exact for all rank-1 torsion-free abelian groups $X$. This paper sets forth the basic theory of quasi-balanced sequences, with particular attention given to the case in which $C$ is a Butler group. The special case where $B$ is almost completely decomposable gives rise to a descending chain of classes of Butler groups. This chain is a generalization of the chain of Kravchenko classes that arise from balanced...