Annona emarginata (Schltdl.) H. Rainer, commonly known as “arachichú”, “araticú”, “aratigú”, and “yerba mora”, is a plant that grows in Argentina. Infusions and decoctions are used in folk medicine as a gargle against throat pain and for calming toothache; another way to use the plant for these purposes is chewing its leaves. Extracts from bark, flowers, leaves, and fruits from A. emarginata were subjected to antibacterial assays against a panel of Gram (+) and Gram (−) pathogenic bacteria according to Clinical and Laboratory Standards Institute protocols. Extracts from the stem bark and leaves showed moderate activity against the bacteria tested with values between 250–1000 µg/mL. Regarding flower extracts, less polar extracts (hexane, dichloromethane) showed very strong antibacterial activity against methicillin-sensitive Staphylococcus aureus ATCC 25923 and methicillin-resistant S. aureus ATCC 43300 with values between 16–125 µg/mL. Additionally, hexane extract showed activity against Klebsiella pneumoniae (MIC = 250 µg/mL). The global methanolic extract of the fruits (MeOHGEF) was also active against the three strains mentioned above, with MICs values 250–500 µg/mL. Bioassay-guided fractionation of MeOHGEF led to the isolation of a new main compound—(R)-2-(4-methylcyclohex-3-en-1-yl)propan-2-yl (E)-3-(4-hydroxyphenyl)acrylate (1). The structure and relative configurations have been determined by means of 1D and 2D NMR techniques, including COSY, HMQC, HMBC, and NOESY correlations. Compound 1 showed strong antimicrobial activity against all Gram (+) species tested (MICs = 3.12–6.25 µg/mL). In addition, the synthesis and antibacterial activity of some compounds structurally related to compound 1 (including four new compounds) are reported. A SAR study for these compounds was performed based on the results obtained by using molecular calculations.