We investigate how non-stabilizer resources enable the emergence of quantum state designs within the projected ensemble. Starting from initial states with finite magic and applying resource-free Clifford circuits to scramble them, we analyze the ensemble generated by performing projective Pauli measurements on a subsystem of the final state. Using both analytical arguments and large-scale numerics, we show that the projected ensemble converges towards a state $k$-design with an error that decays exponentially with the $k$-th Stabilizer Rényi Entropy of the pre-measurement state, via a Magic-Induced Design Ansatz (MIDA) that we introduce. We identify a scaling form that depends on the initial state only through its magic $M_k$, which we corroborate for distinct classes of initial states through numerical simulations and analytical calculations of the frame potential. For finite-depth Clifford unitaries, we show that the timescales at which state designs emerge are controlled by the transport of magic. We identify a "magic teleportation" mechanism whereby non-Clifford resources injected locally spread through Clifford scrambling and measurements across distances beyond the lightcone. Our results demonstrate that an amount of magic scaling with the probed subsystem size, yet independent of the total system size, suffices to generate highly random states, providing a systematic route toward generating quantum state designs in early fault-tolerant devices.