In some applications, perovskite light-emitting diodes (PeLEDs) shall operate in pulsed mode. The generation of high-intensity light pulses requires PeLED driving by high-power electrical pulses, which can lead to deterioration of PeLED performance and their degradation. Contrarily, PeLEDs operating in a nonconventional regime, based on the so-called overshoot effect, enable the generation of short, high-intensity optical pulses at relatively low driving pulse power. Here, the generation of overshoot pulses (OSPs) by FAPI PeLEDs is analyzed. The intensity and shape of the OSPs are determined not only by the driving (injection) pulse parameters but also by the offset voltage applied between the injection pulses and the afterpulse applied after the injection pulse. The offset voltage determines the distribution of the mobile ions, which strongly affect the internal electric field during the pulse action and after its termination, thus strongly affecting the evolution of the conventional electroluminescence (EL) and generation of the OSPs. Meanwhile, the afterpulse voltage controls the intensity and duration of the OSPs. The intensity of the OSPs increases strongly at temperatures below ≈200 K. Mathematical modeling reproduces the EL dynamics and reveals two distinct PeLED operation modes: one that facilitates OSP generation and another that prevents it.