Pulse-coupled threshold units serve as paradigmatic models for a wide range of complex systems. When the state variable of a unit crosses a threshold, the unit sends a pulse that is received by other units, thereby mediating the interactions. At the same time, the state variable of the sending unit is reset. Here we present and analyze a class of pulse-coupled oscillators where the reset may be partial only and is mediated by a partial reset function. Such a partial reset characterizes intrinsic physical or biophysical features of a unit, e.g., resistive coupling between dendrite and soma of compartmental neurons; at the same time the description in terms of a partial reset enables a rigorous mathematical investigation of the collective network dynamics. The partial reset acts as a desynchronization mechanism. For N all-to-all pulse-coupled oscillators an increase in the strength of the partial reset causes a sequence of desynchronizing bifurcations from the fully synchronous state via states with large clusters of synchronized units through states with smaller clusters to complete asynchrony. By considering inter-and intracluster stability we derive sufficient and necessary conditions for the existence and stability of cluster states on the partial reset function and on the intrinsic dynamics of the oscillators. For a specific class of oscillators we obtain a rigorous derivation of all N-1 bifurcation points and demonstrate that already arbitrarily small changes in the reset function may produce the entire sequence of bifurcations. We illustrate that the transition is robust against structural perturbations and prevails in the presence of heterogeneous network connectivity and changes in the intrinsic oscillator dynamics.