For high-dimensional heterogeneous Kuramoto oscillator networks (HDHKONs) evolving in continuous time, achieving global synchronization can be exceedingly challenging, and in many cases, it may even appear impossible for various network configurations, including Platonic solids and Archimedean solids, or large-scale networks. Herein, a pinning impulsive control approach for HDHKONs is developed to attain global synchronization on the unit sphere without imposing constraints on initial phase distributions. With this approach, the entire network is globally stabilized onto an objective trajectory by impulsively controlling only a small fraction of oscillators. Furthermore, several synchronization criteria are provided to ensure that the global synchronization procedure is successful. These criteria are easily verified and shed light on the interplay among network parameters, the percentage of controlled oscillators, impulsive intensity, and impulsive frequency. Finally, two examples are implemented to validate the theoretical results.