The receiver operating characteristic (ROC) curve is a commonly used statistical method to assess the efficacy of a diagnostic test or biomarker measured on a continuous scale. This work presents a versatile approach using a non-uniform rational B-spline (NURBS) for estimating the ROC curve. This approach uses control points, weights, and the knot sequence to more accurately estimate the true ROC curve. The new method applies linear constraints to the NURBS basis function coefficients to smooth the empirical ROC curve and guarantee a non-decreasing function. Moreover, as a specific case, a NURBS curve devoid of interior knots simplifies to the Bernstein polynomial when all weight values are equal. We conduct Monte Carlo simulation studies to evaluate how well the NURBS-based estimator works in different scenarios. We compare our estimator to the empirical ROC, the kernel-based ROC, and Bernstein polynomial estimators in terms of the averaged squared errors. We also apply our method to two real medical datasets, such as metastatic kidney cancer and diffuse large B-cell lymphoma datasets. According to the findings from both the real and simulated data, the NURBS method is a powerful alternative for estimating the ROC curve.