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Nematic phases, where electrons in a solid spontaneously break rotational symmetry while preserving the translational symmetry, exist in several families of unconventional superconductors [1, 2]. Although superconductivity mediated by nematic fluctuations is well established theoretically [3-7], it has yet to be unambiguously identified experimentally [8, 9]. A major challenge is that nematicity is often intertwined with other degrees of freedom, such as magnetism and charge order. The FeSe$_{1-x}$S$_x$ family of iron based superconductors provides a unique opportunity to explore this concept, as it features an isolated nematic phase that can be suppressed by sulfur substitution at a quantum critical point (QCP) near $x_c = 0.17$, where nematic fluctuations are the largest [10-12]. Here, we performed scanning tunneling spectroscopy measurements to visualize Boguliubov quasiparticle interference patterns, from which we determined the momentum structure of the superconducting gap near the Brillouin zone $\Gamma$ point of FeSe$_{0.81}$S$_{0.19}$. The results reveal an anisotropic, near nodal gap with minima that are $45^\circ$ rotated with respect to the Fe-Fe direction, characteristic of a nematic pairing interaction, contrary to the usual isotropic gaps due to spin mediated pairing in other tetragonal Fe-based superconductors. The results are also in contrast with pristine FeSe, where the pairing is mediated by spin fluctuations and the gap minima are aligned with the Fe-Fe direction. Therefore, the measured gap structure demonstrates not only a fundamental change of the pairing mechanism across the phase diagram of FeSe$_{1-x}$S$_x$, but it also indicates the existence of superconductivity mediated by nematic fluctuations in FeSe$_{0.81}$S$_{0.19}$.