Novel isoxazole–triazole conjugates have been efficiently synthesized using 3-formylchromone as starting material according to a multi-step synthetic approach. The structures of the target conjugates and intermediate products were characterized by standard spectroscopic techniques (¹H NMR and ¹³C NMR) and confirmed by mass spectrometry (MS). The all-synthesized compounds were screened for their antibacterial activity against three ATCC reference strains, namely Staphylococcus aureus ATCC 25923, Staphylococcus aureus ATCC BAA-44, and Escherichia coli ATCC 25922 as well as one strain isolated from the hospital environment Pseudomonas aeruginosa. The findings indicate that conjugate 7b exhibits a stronger antibacterial response against the tested Escherichia coli ATCC 25922 and Pseudomonas aeruginosa pathogenic strains compared to the standard antibiotics. Furthermore, hybrid compound 7b proved to have a bactericidal action on the Escherichia coli ATCC 25922 strain, as evidenced by the results of the MBC determination. Moreover, the ADMET pharmacokinetic characteristics revealed a favorable profile for the examined compound, as well as a good level of oral bioavailability. Molecular docking and molecular dynamics simulations were performed to explore the inhibition mechanism and binding energies of conjugate 7b with the proteins of Escherichia coli and Pseudomonas aeruginosa bacterial strains. The in silico results corroborated the data observed in the in vitro evaluation for compound 7b.