Due to the growing popularity of herbal extract-loaded hydrogels, this study assessed the biological activity of extracts and hydrogels containing three types (water (WE), water–ethanol (EE) and water–glycerin (GE)) of Cornus mas L. (dogwood) extracts. The content of biologically active compounds in the extracts was assessed using the UPLC-DAD-MS technique. Antioxidant properties were assessed by using DPPH and ABTS radicals and measuring the intracellular level of reactive oxygen species. Alamar Blue and Neutral Red tests were used to measure the cytotoxicity of the tested samples on skin cells—fibroblasts and keratinocytes. Cell migration and the anti-aging activity of the tested extracts and hydrogels were assessed. Transepidermal water loss and skin hydration after applying the hydrogels to the skin were also determined. A chromatographic analysis revealed that the extracts contained polyphenols, including gallic, caftaric, protocatechuic, chlorogenic, ellagic and p-coumaroylquinic acids, as well as iridoids, with loganic acid as the predominant component. Additionally, they contained cyanidin 3-O-galactoside, pelargonidin 3-O-glucoside and quinic acid. The obtained results show that the tested extracts and hydrogels had strong antioxidant properties and had a positive effect on the viability of skin cells in vitro. Additionally, it was shown that they stimulated the migration of these cells and had the ability to inhibit the activity of collagenase and elastase. Moreover, the tested hydrogels increased skin hydration and prevented transepidermal water loss. The obtained results indicate that the developed hydrogels may be effective delivery systems for phytochemicals contained in dogwood extracts.