Sunflower oil industry is interested in both increasing yield and preserving quality of the extracted oil. The current work aimed to investigate the effect of texturing by instant controlled pressure-drop (DIC) on oil cold-press extraction. DIC texturing consists of heating at saturated dry-steam pressure P (200–700 kPa or 110–160 °C), for a short processing time t, followed by an instantaneous pressure-drop (20–40 ms) towards a vacuum (3–5 kPa). This resulted in increasing oil pressing-yield up to 46% and 33%, for linoleic and oleic varieties, respectively. It also increased the whole-oil availability by 11%, with the conservation of the biochemical composition of both oil and residual cake. A multi-criteria optimization of the DIC operating parameters; P and t, was performed through a 5-level Central Composite Design (CCD) and statistical analyses. This highlighted the direct significant effect of t on the oil yield while greatly preserving the quality of the final products.