We introduce a simple model of self-propelled particles connected by linear springs that describes a semi-rigid formation of active agents without explicit alignment rules. The model displays a discontinuous transition at a critical noise level, below which the group self-organizes into a collectively translating or rotating state. We identify a novel elasticity-based mechanism that cascades self-propulsion energy towards lower-energy modes as responsible for such collective motion and illustrate it by computing the spectral decomposition of the elastic energy. We study the model's convergence dynamics as a function of system size and derive analytical stability conditions for the translating state in a continuous elastic sheet approximation. We explore the dynamics of a ring-shaped configuration and of local angular perturbations of an aligned state. We show that the elasticity-based mechanism achieves collective motion even in cases with heterogeneous self-propulsion speeds. Given its robustness, simplicity and ubiquity, this mechanism could play a relevant role in various biological and artificial swarms.