The complete separation of sister chromatids during anaphase is a fundamental requirement for successful mitosis. Therefore, divisions with either persistent DNA-based connections or lagging chromosome fragments threaten aneuploidy if unresolved. Here, we demonstrate the existence of an anaphase mechanism in normally dividing cells in which pervasive connections between telomeres of segregating chromosomes aid in rescuing lagging chromosome fragments. We observe that in a large proportion of Drosophila melanogaster neuronal stem cell divisions, early anaphase sister and non-sister chromatids remain connected by thin telomeric DNA threads. Normally, these threads are resolved in mid-to-late anaphase via a spatial mechanism. However, we find that the presence of a nearby unrepaired DNA break recruits histones, BubR1 kinase, Polo kinase, Aurora B kinase, and BAF to the telomeric thread of the broken chromosome, stabilizing it. Stabilized connections then aid lagging chromosome rescue. These results suggest a model in which pervasive anaphase telomere-telomere connections that are normally resolved quickly can instead be stabilized to retain wayward chromosome fragments. Thus, the liability of persistent anaphase inter-chromosomal connections in normal divisions may be offset by their ability to maintain euploidy in the face of chromosome damage and genome loss.