The effect of aging (at 773 K for 1 h) on the cyclic stability of superelasticity was studied in preliminarily quenched (after annealing at 1448 K for 1 h) Ni44Fe19Ga27Co10 [001]-oriented single crystals. It was shown that NiFeGaCo single crystals (both quenched and aged) exhibited highly stable superelasticity during more than 105 loading/unloading cycles in compression without destruction. The degradation of the superelasticity curves was divided into stages, and each stage of degradation was carefully considered. The precipitation of ω-phase particles led to a change in the degradation mechanism of superelasticity in the aged crystals. The main degradation mechanism for quenched crystals is a formation of uniform distribution of dislocations; for aged crystals, the microstructural degradation mechanisms consist of a non-uniform distribution of dislocations around the particles and the appearance of residual martensite.