The design of biomaterial platforms able to release bioactive molecules is mandatory in tissue repair and regenerative medicine. In this context, electrospinning is a user-friendly, versatile and low-cost technique, able to process different kinds of materials in micro- and nano-fibers with a large surface area-to-volume ratio for an optimal release of gaseous signaling molecules. Recently, the antioxidant and anti-inflammatory properties of the endogenous gasotramsmitter hydrogen sulfide (H2S), as well as its ability to stimulate relevant biochemical processes on the growth of mesenchymal stem cells (MSC), have been investigated. Therefore, in this work, new poly(lactic) acid fibrous membranes (PFM), doped and functionalized with H2S slow-releasing donors extracted from garlic, were synthetized. These innovative H2S-releasing mats were characterized for their morphological, thermal, mechanical, and biological properties. Their antimicrobial activity and effects on the in vitro human cardiac MSC growth, either in the presence or in the absence of oxidative stress, were here assessed. On the basis of the results here presented, these new H2S-releasing PFM could represent promising and low-cost scaffolds or patches for biomedical applications in tissue repair.