The fundamental phenomenon of Bose-Einstein condensation has been observed in different systems of real particles and quasiparticles. The condensation of real particles is achieved through a major reduction in temperature, while for quasiparticles, a mechanism of external injection of bosons by irradiation is required. Here, we present a new and universal approach to enable Bose-Einstein condensation of quasiparticles and to corroborate it experimentally by using magnons as the Bose-particle model system. The critical point to this approach is the introduction of a disequilibrium of magnons with the phonon bath. After heating to an elevated temperature, a sudden decrease in the temperature of the phonons, which is approximately instant on the time scales of the magnon system, results in a large excess of incoherent magnons. The consequent spectral redistribution of these magnons triggers the Bose-Einstein condensation. A new method to form Bose-Einstein condensates of quasiparticles based on the rapid decrease in the phonon temperature was proposed and shown experimentally.