Shape memory effect, the ability to recover a pre-deformed shape on heating, results from a reversible martensitic transformation between austenite and martensite phases. Here, we demonstrate a strategy of designing high-entropy alloys (HEAs) with high-temperature shape memory effect in the CrMnFeCoNi alloy system. First, we calculate the difference in Gibbs free energy between face-centered-cubic (FCC) and hexagonal-close-packed (HCP) phases, and find a substantial increase in thermodynamic equilibrium temperature between the FCC and HCP phases through composition tuning, leading to thermally- and stress-induced martensitic transformations. As a consequence, the shape recovery temperature in non-equiatomic CrMnFeCoNi alloys can be increased to 698 K, which is much higher than that of conventional shape memory alloys (SMAs) and comparable to that of B2-based multi-component SMAs containing noble metals (Pd, Pt, etc.) or refractory metals (Zr, Hf, etc.). This result opens a vast field of applications of HEAs as a novel class of cost-effective high-temperature SMAs.