Over the course of operation, asphalt road pavements are subjected to damage from car traffic loads and environmental factors. One of the possible methods of strengthening damaged asphalt pavements may be the application of an additional rigid layer in the form of a cement concrete slab with continuous reinforcement. This paper presents a material-technological and structural solution for composite pavement where a cement concrete slab with continuous HFRP bar reinforcement is used for strengthening. Based on laboratory tests, the serviceability of composite bar reinforcement of rigid pavement slabs was shown. A design for strengthening asphalt pavement with a concrete slab with steel bar and corresponding HFRP bar reinforcement was developed. The composition of a pavement cement concrete mix was designed, and experimental sections were formed. Based on laboratory tests of samples collected from the surfaces of experimental sections and the diagnostic tests carried out in “in situ” conditions, the authors will try, in the nearest future (Part II: In situ observations and tests), to confirm the effectiveness of strengthening asphalt pavements with cement concrete slabs with HFRP components.