Biofilms currently represent the most prevalent bacterial lifestyle, enabling them to resist environmental stress and antibacterial drugs. Natural antibacterial agents could be a safe solution for controlling bacterial biofilms in food industries without affecting human health and environmental safety. A methanolic extract of Azadirachta indica (neem) leaves was prepared and analyzed using gas chromatography–mass spectrometry for the identification of its phytochemical constituents. Four food-borne bacterial pathogens (Bacillus cereus, Novosphingobium aromaticivorans, Klebsiella pneumoniae, and Serratia marcescens) were tested for biofilm formation qualitatively and quantitatively. The antibacterial and antibiofilm properties of the extract were estimated using liquid cultures and a microtiter plate assay. The biofilm inhibition mechanisms were investigated using a light microscope and molecular docking technique. The methanolic extract contained 45 identified compounds, including fatty acids, ester, phenols, flavonoids, terpenes, steroids, and antioxidants with antimicrobial, anticancer, and anti-inflammatory properties. Substantial antibacterial activity in relation to the extract was recorded, especially at 100 μg/mL against K. pneumoniae and S. marcescens. The extract inhibited biofilm formation at 100 μg/mL by 83.83% (S. marcescens), 73.12% (K. pneumoniae), and 54.4% (N. aromaticivorans). The results indicate efficient biofilm formation by the Gram-negative bacteria S. marcescens, K. pneumoniae, and N. aromaticivorans, giving 0.74, 0.292, and 0.219 OD at 595 nm, respectively, while B. cereus was found to have a low biofilm formation potential, i.e., 0.14 OD at 595 nm. The light microscope technique shows the antibiofilm activities with the biofilm almost disappearing at 75 μg/mL and 100 μg/mL concentrations. This antibiofilm property was attributed to DNA gyrase inhibition as illustrated by the molecular docking approach.