Background/Objectives: Wounds from burns are susceptible to infections, allowing multidrug-resistant microorganisms to complicate treatments and patient recovery. This highlights the development of new strategies to control these microorganisms. This work evaluated the antibacterial activity of hydrogels containing biogenic silver nanoparticles (bio-AgNP) and Origanum vulgare essential oil (OEO) against multidrug-resistant bacteria. Methods: The formulations were subjected to organoleptic, pharmacotechnical, and stability characterization and antimicrobial activity assessment by time–kill tests and alternative methods, an ex vivo model using porcine skin, and an in vivo model using Galleria mellonella. Results: All hydrogels maintained their stability after the thermal stress. The hydrogel containing bio-AgNP + OEO 1% (HAgNP + OEO1) presented bactericidal effectiveness, within 2 h, against both Gram-positive and Gram-negative multidrug-resistant bacteria in the time–kill test. For alternative testing, HAgNP + OEO1 was compared with 1% silver sulfadiazine (SS) and the base formulation. In the ex vivo test, both HAgNP + OEO1 and SS treatments showed a similar reduction in superficial washing of the burn for S. aureus 999, while for P. aeruginosa, the reduction was more expressive for SS treatment. In the burn tissue, HAgNP + OEO1 treatment was more effective against S. aureus 999, while for P. aeruginosa 1461, both formulations were similarly effective. In the Galleria mellonella test, survival rates after 48 h were 84% for the control group (base) and 50% for both HAgNP + OEO1 and SS treatment groups. Conclusions: This study demonstrates that the hydrogel combining antimicrobials is effective against multidrug-resistant microorganisms, offering a promising alternative for the treatment of infected burns.