Stress-associated protein (SAP) genes—encoding A20/AN1 zinc-finger domain-containing proteins—play pivotal roles in regulating stress responses, growth, and development in plants. They are considered suitable candidates to improve abiotic stress tolerance in plants. However, the SAP gene family in sweetpotato (Ipomoea batatas) and its relatives is yet to be investigated. In this study, 20 SAPs in sweetpotato, and 23 and 26 SAPs in its wild diploid relatives Ipomoea triloba and Ipomoea trifida were identified. The chromosome locations, gene structures, protein physiological properties, conserved domains, and phylogenetic relationships of these SAPs were analyzed systematically. Binding motif analysis of IbSAPs indicated that hormone and stress responsive cis-acting elements were distributed in their promoters. RT-qPCR or RNA-seq data revealed that the expression patterns of IbSAP, ItbSAP, and ItfSAP genes varied in different organs and responded to salinity, drought, or ABA (abscisic acid) treatments differently. Moreover, we found that IbSAP16 driven by the 35 S promoter conferred salinity tolerance in transgenic Arabidopsis. These results provided a genome-wide characterization of SAP genes in sweetpotato and its two relatives and suggested that IbSAP16 is involved in salinity stress responses. Our research laid the groundwork for studying SAP-mediated stress response mechanisms in sweetpotato.