Abasic sites are one of the most frequent forms of DNA damage that interfere with DNA replication. However, abasic sites exhibit complex effects because they can be processed into other types of DNA damage. Thus, it remains poorly understood how abasic sites affect replisome progression, which replication-coupled repair pathways they elicit, and whether this is affected by the template strand that is damaged. Using Xenopus egg extracts, we developed an approach to analyze replication of DNA containing a site-specific, stable abasic site on the leading or lagging strand template. We show that abasic sites robustly stall synthesis of nascent DNA strands but exert different effects when encountered on the leading or lagging strand template. At a leading strand AP site, replisomes stall ∼100 bp from the lesion until it is bypassed or a converging fork triggers termination. At a lagging strand abasic site, replisome progression is unaffected and lagging strands are reprimed downstream, generating a post-replicative gap, which is then bypassed. Despite different effects on replisome progression, both leading and lagging strand abasic sites rely on translesion DNA synthesis for bypass. Our results detail similarities and differences between how leading and lagging strand AP sites affect vertebrate DNA replication.