Over the past decades, the problem of bacterial resistance to most antibiotics has become a serious threat to patients’ survival. Nevertheless, antibiotics of a novel class have not been approved since the 1980s. The development of antibiotic potentiators is an appealing alternative to the challenging process of searching for new antimicrobials. Production of H₂S—one of the leading defense mechanisms crucial for bacterial survival—can be influenced by the inhibition of relevant enzymes: bacterial cystathionine γ-lyase (bCSE), bacterial cystathionine β-synthase (bCBS), or 3-mercaptopyruvate sulfurtransferase (MST). The first one makes the main contribution to H₂S generation. Herein, we present data on the synthesis, in silico analyses, and enzymatic and microbiological assays of novel bCSE inhibitors. Combined molecular docking and molecular dynamics analyses revealed a novel binding mode of these ligands to bCSE. Lead compound 2a manifested strong potentiating activity when applied in combination with some commonly used antibiotics against multidrug-resistant Acinetobacter baumannii, Pseudomonas aeruginosa, and methicillin-resistant Staphylococcus aureus. The compound was found to have favorable in vitro absorption, distribution, metabolism, excretion, and toxicity parameters. The high effectiveness and safety of compound 2a makes it a promising candidate for enhancing the activity of antibiotics against high-priority pathogens.