Zeus21 (Zippy Early-Universe Solver for 21-cm) captures the nonlocal and nonlinear physics of cosmic dawn to create an effective model for the 21-cm power spectrum and global signal. The code takes advantage of the approximate log-normality of the star-formation rate density (SFRD) during cosmic dawn to compute the 21-cm power spectrum analytically. It agrees with more expensive semi-numerical simulations to roughly 10% precision, but has comparably negligible computational cost (~ s) and memory requirements. Zeus21 pairs well with data from HERA, but can be used for any 21-cm inference or prediction. Its capabilities include finding the 21-cm power spectrum (at a broad range of k and z), the global signal, IGM temperatures (Tk, Ts, Tcolor), neutral fraction xHI, Lyman-alpha fluxes, and the evolution of the SFRD; all across cosmic dawn z=5-35. It can also predict UVLFs for HST and JWST. Zeus21 can use three different astrophysical models, one of which emulates 21cmFAST (ascl:1102.023), and can vary the cosmology through CLASS (ascl:1106.020).