The effects of a single-screw extruder configuration and processing variables such as conventional extrusion or hybrid treatments with xylanase were tested on the extrusion performance and selected characteristics of the developed non-starch polysaccharide-rich (NSP-rich) wheat flour. L/D 16 and 20 extruder configurations with various screw profiles were used. The interactions between processing variables (moisture content 23, 25, 27%; screw speed 40, 60, 80 rpm; xylanase level 0, 50, 100 ppm) were assessed to indicate energy consumption and the rheological properties of flour. The results showed that the possibility of obtaining enzyme-assisted extruded flour products derived from flours of varying characteristics depended on the processing conditions. The application of various extruder configurations and screw profiles showed significant effects on both processing behavior and rheological characteristics. The longer L/D 20 extruder configuration using a screw profile with mixing elements allowed us to obtain products with lower extrusion pressure (max. 20.8 bar) and energy requirements (max. SME = 33.1 kWh/kg) and better rheological properties (max. Hyd = 69.2%, less intensive starch gelatinization with max. C3 = 1.47 Nm) than the L/D 16 version. The extruded wheat flour was characterized by improved hydration properties and limited retrogradation tendency, especially when hybrid extrusion with xylanase was applied. This may lead to favorable results, as the newly developed enzymatic extrusion modification method produces NSP-rich wheat flour with specific techno-functional and rheological characteristics that can be seen as a potential “clean label” enhancer in bakery products. Our statistical analysis confirmed feed moisture and screw speed to be the variables with the most significant effect on wheat flour features.