CRISPR–Cas9-mediated homology-directed repair (HDR) can introduce desired mutations at targeted genomic sites, but achieving high efficiencies is a major hurdle in many cell types, including cells deficient in DNA repair activity. In this study, we used genome-wide screening in Fanconi anemia patient lymphoblastic cell lines to uncover suppressors of CRISPR–Cas9-mediated HDR. We found that a single exonuclease, TREX1, reduces HDR efficiency when the repair template is a single-stranded or linearized double-stranded DNA. TREX1 expression serves as a biomarker for CRISPR–Cas9-mediated HDR in that the high TREX1 expression present in many different cell types (such as U2OS, Jurkat, MDA-MB-231 and primary T cells as well as hematopoietic stem and progenitor cells) predicts poor HDR. Here we demonstrate rescue of HDR efficiency (ranging from two-fold to eight-fold improvement) either by TREX1 knockout or by the use of single-stranded DNA templates chemically protected from TREX1 activity. Our data explain why some cell types are easier to edit than others and indicate routes for increasing CRISPR–Cas9-mediated HDR in TREX1-expressing contexts.