Background/Objectives: Burn injuries represent a global public health concern, accounting for approximately 265,000 deaths annually and often leading to severe infections. With the increasing prevalence of multidrug-resistant (MDR) bacteria, innovative therapeutic strategies such as nanoparticle-based topical formulations have gained attention. This study proposed the development of a hydrogel for burn treatment containing biogenic silver nanoparticles (BioAgNPs), hyaluronic acid (HA), and allantoin (AL). Methods: BioAgNPs were previously characterized by transmission electron microscopy (TEM) and incorporated into a hydrogel containing HA and AL, which was physicochemically characterized by pH, spreadability, and energy-dispersive X-ray spectroscopy (EDX). Antibacterial activity was evaluated by broth microdilution, agar diffusion, and time–kill assays against standard and MDR bacterial strains. Cytotoxicity was assessed using the MTT assay in L929 cells, and wound-healing potential was investigated through an in vitro scratch assay to evaluate cell migration and proliferation. Results: BioAgNPs exhibited antibacterial activity against reference strains and MDR isolates, determining the minimum inhibitory concentrations (MIC) and minimum bactericidal concentrations (MBC). HA and AL were non-toxic, while BioAgNPs demonstrated low cytotoxic activity. Although HA and AL did not exhibit antibacterial properties, they promoted cell migration and proliferation. The formulation exhibited physicochemical and pharmaceutical stability, showing suitable properties for topical use, and presented significant antimicrobial action, with bacterial elimination occurring within 2 h of contact, except for S. aureus. Conclusions: Thus, the hydrogel presents a promising alternative for the topical treatment of infected burns, with potential application in combating multidrug-resistant bacteria, being able to eliminate MDR Acinetobacter baumannii.