We present a high-order surface quadrature (HOSQ) for accurately approximating regular surface integrals on closed surfaces. The initial step of our approach rests on exploiting square-squeezing–a bilinear square-simplex homeomorphism, re-parametrizing any surface triangulation to a quadrilateral mesh. For each resulting quadrilateral domain we interpolate the geometry by tensor polynomials in Chebyshev–Lobatto grids, circumventing the Runge phenomenon while leveraging the capabilities of spectral differentiation. Posterior the tensor-product Clenshaw-Curtis quadrature is applied to compute the resulting integral. We demonstrate efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries.