This study aimed to investigate the impact of chicory root addition (20–40%) and extrusion conditions (moisture content from 16.3 to 22.5%, and screw speed from 500 to 900 rpm) on bioactive compounds content (inulin, sesquiterpene lactones, and polyphenols) of gluten-free rice snacks. Chicory root is considered a potential carrier of food bioactives, while extrusion may produce a wide range of functional snack products. The mineral profiles were determined in all of the obtained extrudates in terms of Na, K, Ca, Mg, Fe, Mn, Zn, and Cu contents, while antioxidative activity was established through reducing capacity, DPPH (2,2-diphenyl-1-picrylhydrazyl) and ABTS (2,2-azino-bis(3-ethylbenzothiazoline-6-sulfonic acid) tests. Chicory root addition contributed to the improvement of bioactive compounds and mineral contents, as well as antioxidative activities in all of the investigated extrudates in comparison to the pure-rice control sample. An increase in moisture content raised sesquiterpene lactones and minerals, while high screw speeds positively affected polyphenols content. The achieved results showed the important impact of the extrusion conditions on the investigated parameters and promoted chicory root as an attractive food ingredient in gluten-free snack products with high bioactive value.