In this work, three cyanoalkyl silicone GC stationary-phase polymers, namely OV-105, OV-225, and OV-275, were investigated as potential extractants for dispersive liquid–liquid microextraction (DLLME). The OV-225 polymer (cyanopropylmethyl-phenylmethylsilicone) exhibited the cleanest chromatographic background and was extensively studied. The proposed polymer was tested through the DLLME of four non-steroidal anti-inflammatory drugs from aqueous samples, followed by HPLC separation with UV detection at 230 nm. To achieve the maximum enrichment, the experimental conditions that influence the DLLME process were optimized using one-factor-at-a-time and design-of-experiment (DoE) approaches. The extraction variables (polymer mass, dispersive solvent volume, buffer pH, and mixing time) were screened by implementing a two-level full factorial design (FFD). Significant variables were fine-tuned using response surface methodology based on a face-centered central composite design (CCD). The optimum conditions were 10 mg of polymer (extraction medium); 50 µL of tetrahydrofuran (dispersive solvent); 100 µL of phosphate buffer pH 2.75 ([PO43−] = 100 mM); and 3 min of vortex mixing. The addition of salt had a minimal effect on the enrichment factors. In the optimum conditions, enrichment factors up to 46 were achieved using 1.5 mL samples. Calibration curves exhibited correlation coefficients > 0.999 using 4-pentylbenzoic acid as an internal standard. The limits of quantitation were 5 ng/mL for naproxen, 10 ng/mL for diflunisal, 25 ng/mL for indomethacin, and 75 ng/mL for ibuprofen. The analysis of spiked tap water samples showed adequate relative recoveries and precision. In conclusion, the proposed polymer (OV-225) is a potential greener alternative to traditional organic extractants used in DLLME.