A new type of gluten-free pasta has been developed based on a rice-buckwheat mixture. The aim of the study was to investigate the effect of process parameters of moisture content (30, 33, and 36%), barrel temperature (80, 100, and 120 °C), and screw speed (60, 80, and 100 rpm) on cooking and textural properties of rice-buckwheat pasta produced by a single-screw extrusion-cooker. The process uses response surface methodology based on a Box-Behnken experimental design. Results showed that with regard to this rice-buckwheat pasta, raising moisture content of the raw materials increased cooking loss and stickiness, but decreased firmness, while increasing barrel temperature reduced cooking loss and stickiness, but increased hardness and firmness. Screw speed increase also affected positively hardness and firmness of the obtained products. Thus, optimal conditions (moisture content 30%, barrel temperature 120 °C, and screw speed 80 rpm) were established to produce good quality rice-buckwheat pasta. At this optimum, the pasta showed a compact and homogeneous inside microstructure. Furthermore, the pasta products exhibited low cooking loss (less than 6%), good hardness and firmness, with low stickiness and acceptable scores for all sensory attributes and for overall quality.