Functional foods represent a new and thriving area of research. A significant direction of these studies is based on new products containing edible house cricket-derived additives. The aim of the presented studies was to determine the effect of using cricket powder (at 10% and 30% content) on the extrusion cooking parameters and the nutritional value, antioxidant activity, and selected physical properties of extruded potato-based snack pellets. The results suggest that house cricket powder is a promising functional ingredient. The processing efficiency and the physical and functional properties of the extrudates, including SME, WAI, WSI, bulk density, and mechanical durability, were affected by the addition of cricket powder, screw speed, and moisture content. Generally, higher levels of cricket powder reduced processing efficiency and altered structural properties due to changes in composition, particularly the balance between protein, fiber, and starch. The addition of cricket powder significantly improved antioxidant activity (>94% of DPPH scavenging for 30% content of additive) and increased the total polyphenol content in the assessed samples in comparison to potato bases (212.3 and 21.7 μg GAE/g dry weight, respectively). These innovative snack pellets containing cricket powder could be an appealing option due to their potential health benefits.