In the biomedical sector the availability of engineered scaffolds and dressings that control and reduce inflammatory states is highly desired, particularly for the management of burn wounds. In this work, we demonstrate for the first time, to the best of our knowledge, that electrospun fibrous dressings of poly(octyl cyanoacrylate) (POCA) combined with polypropylene fumarate (PPF) possess anti-inflammatory activity and promote the fast and effective healing of mild skin burns in an animal model. The fibers produced had an average diameter of (0.8 ± 0.1) µm and they were able to provide a conformal coverage of the injured tissue. The application of the fibrous mats on the burned tissue effectively reduced around 80% of the levels of pro-inflammatory cytokines in the first 48 h in comparison with un-treated animals, and enhanced skin epithelialization. From histological analysis, the skin thickness of the animals treated with POCA : PPF dressings appeared similar to that of one of the naïve animals: (13.7 ± 1.4) µm and (14.3 ± 2.5) µm for naïve and treated animals, respectively. The density of dermal cells was comparable as well: (1100 ± 112) cells mm−2 and (1358 ± 255) cells mm−2 for naïve and treated mice, respectively. The results demonstrate the suitability of the electrospun dressings in accelerating and effectively promoting the burn healing process.