APOBEC cytidine deaminases mutate cancer genomes by converting cytidines into uridines within ssDNA during replication. Although uracil DNA glycosylases limit APOBEC-induced mutation, it is unknown if subsequent base excision repair (BER) steps function on replication-associated ssDNA. Hence, we measured APOBEC3B-induced CAN1 mutation frequencies in yeast deficient in BER endonucleases or DNA damage tolerance proteins. Strains lacking Apn1, Apn2, Ntg1, Ntg2 or Rev3 displayed wild-type frequencies of APOBEC3B-induced canavanine resistance (CanR). However, strains without error-free lesion bypass proteins Ubc13, Mms2 and Mph1 displayed respective 4.9-, 2.8- and 7.8-fold higher frequency of APOBEC3B-induced CanR. These results indicate that mutations resulting from APOBEC activity are avoided by deoxyuridine conversion to abasic sites ahead of nascent lagging strand DNA synthesis and subsequent bypass by error-free template switching. We found this mechanism also functions during telomere re-synthesis, but with a diminished requirement for Ubc13. Interestingly, reduction of G to C substitutions in Ubc13-deficient strains uncovered a previously unknown role of Ubc13 in controlling the activity of the translesion synthesis polymerase, Rev1. Our results highlight a novel mechanism for error-free bypass of deoxyuridines generated within ssDNA and suggest that the APOBEC mutation signature observed in cancer genomes may under-represent the genomic damage these enzymes induce.