Conserved DNA-damage responses (DDRs) sense genome damage and prevent mitosis of broken chromosomes. How cells lacking DDRs cope with broken chromosomes during mitosis is poorly understood. DDRs are frequently inactivated in cells with extra genomes (polyploidy), suggesting that study of polyploidy can reveal how cells with impaired DDRs/genome damage continue dividing. Here, we show that continued division and normal organ development occurs in polyploid, DDR-impaired Drosophila papillar cells. As papillar cells become polyploid, they naturally accumulate broken acentric chromosomes but do not apoptose/arrest the cell cycle. To survive mitosis with acentric chromosomes, papillar cells require Fanconi anemia proteins FANCD2 and FANCI, as well as Blm helicase, but not canonical DDR signaling. FANCD2 acts independently of previous S phases to promote alignment and segregation of acentric DNA produced by double-strand breaks, thus avoiding micronuclei and organ malformation. Because polyploidy and impaired DDRs can promote cancer, our findings provide insight into disease-relevant DNA-damage tolerance mechanisms.