Effective management of chronic wounds with excessive exudate may be challenging for medical doctors. Over the years, there has been an increasing interest in the engineering of biomaterials, focusing on the development of polymer-based wound dressings to accelerate the healing of exuding wounds. The aim of this study was to use curdlan, which is known to support wound healing, as a base for the production of superabsorbent hybrid biomaterials (curdlan/agarose and curdlan/chitosan) with the intended use as wound dressings for highly exuding wound management. To evaluate the biomedical potential of the fabricated curdlan-based biomaterials, they were subjected to a comprehensive assessment of their microstructural, physicochemical, and biological properties. The obtained results showed that foam-like biomaterials with highly porous structure (66–77%) transform into soft gel after contact with the wound fluid, acting as typical hydrocolloid dressings. Novel biomaterials have the superabsorbent ability (1 g of the biomaterial absorbs approx. 15 ml of exudate) with horizontal wicking direction while keeping dry edges, and show water vapor transmission rate of approx. 1700–1800 g/m2/day which is recommended for optimal wound healing. Moreover, they are stable in the presence of collagenases, but prone to biodegradation in lysozyme solution (simulated infected wound environment). Importantly, the developed biomaterials are non-toxic and their surface hinders fibroblast attachment, which is essential during dressing changes to avoid damage to newly formed tissues in the wound bed. All mentioned features make the developed biomaterials promising candidates to be used as the wound dressings for the management of chronic wounds with moderate to high exudate.