Obtaining active extracts from beet root leaves and stems (BLS) is an alternative for the valorization of this agricultural waste. Subcritical water extraction (SWE) at 150 °C and 170 °C has been used to obtain these extracts, which were incorporated (6% wt.) into polymer matrices to produce antioxidant films of thermoplastic starch (TPS) and polylactic acid (PLA) for the preservation of sunflower oil. A high extraction yield (67–60% solubilized solids) was achieved, and the extracts contained high levels of total phenols (51–73 mg GAE·g⁻¹ extract) and betalains and great radical scavenging capacity (EC₅₀: 30–22 mg mg⁻¹ DPPH). The highest temperature promoted the extract’s phenolic richness and antioxidant capacity. The TPS and PLA films containing extracts exhibited color and UV-light blocking effects. The extracts reduced the oxygen permeability (OP) and water vapor permeability of PLA films while promoting those of the TPS films. The capacity of the films to preserve sunflower oil from oxidation was mainly controlled by the OP values of the films, which were very high in TPS films with low OP values. However, in the PLA films (which were more permeable to oxygen), the antioxidant extracts provided significant protection against sunflower oil oxidation.