Hyperspherical elliptic (HSE) harmonics are the eigenfunctions of the generalized angular momentum operator obtained by separating variables in HSE coordinates. These functions depend on accessory parameters characterizing the HSE coordinate system and present a more flexible basis on a hypersphere as compared with more familiar hyperspherical polar harmonics. We discuss a special set of HSE harmonics arising in hyperspherical treatments of the three-body problem in the HSE coordinate system introduced in an earlier paper [Tolstikhin et al., Phys. Rev. Lett. 74, 3573 (1995)]. The separation of variables in these coordinates leads to the Heun equation, which is a generalization of the Gauss hypergeometric equation. We develop an efficient method to solve the corresponding one-dimensional eigenvalue problem and thus construct the HSE harmonics, which opens a way for their application in the studies of various atomic, molecular, and nuclear three-body systems.