We investigate the cosmological implications of a phenomenologically relevant subclass of non-polynomial quasi-topological gravity (NPQTG), in which the background dynamics is encoded in a single function of the Hubble parameter. This framework provides a minimal and theoretically consistent extension of general relativity, incorporating higher-curvature effects while preserving second-order field equations and avoiding higher-derivative instabilities. We first establish the general conditions for cosmological viability and construct explicit realizations, including polynomial, quartic, power-law and non-polynomial models, demonstrating how different functional forms lead to distinct expansion histories. Focusing on the quartic and power-law cases, we investigate their late-time cosmological evolution, showing that they exhibit a matter-dominated epoch followed by a dark-energy-dominated epoch, and give rise to an effective dark-energy sector of geometric origin, with dynamical equation-of-state behavior that can lie in the quintessence or phantom regime. We then confront the models with observational data from Type Ia Supernovae, Cosmic Chronometers, and Baryon Acoustic Oscillations, using a Bayesian MCMC analysis. We find that both models are fully compatible with the considered data and reproduce the late-time expansion history with a quality of fit comparable to that of $\Lambda$CDM, although the current observations do not provide statistically significant evidence for a departure from the concordance scenario. Our results demonstrate that NPQTG offers a simple and efficient framework for describing late-time cosmic acceleration with dynamical dark energy, while maintaining theoretical consistency and observational viability.