We introduce an elasticity-based mechanism that drives active particles to self-organize by cascading self-propulsion energy towards lower-energy modes. We illustrate it on a simple model of self-propelled agents linked by linear springs that reach a collectively rotating or translating state without requiring aligning interactions. We develop an active elastic sheet theory, complementary to the prevailing active fluid theories, and find analytical stability conditions for the ordered state. Given its ubiquity, this mechanism could play a relevant role in various natural and artificial swarms.