We introduce COBRA (Cosmology with Optimally factorized Bases for Rapid Approximation), a novel framework for rapid computation of large-scale structure observables. COBRA separates scale dependence from cosmological parameters in the linear matter power spectrum while also minimising the number of necessary basis terms, thus enabling direct and efficient computation of derived and nonlinear observables. Moreover, the dependence on cosmological parameters is efficiently approximated using radial basis function interpolation. We apply our framework to decompose the linear matter power spectrum in the standard LCDM scenario, as well as by adding curvature, dynamical dark energy and massive neutrinos, covering all redshifts relevant for Stage IV surveys. With only a dozen basis terms, COBRA reproduces exact Boltzmann solver calculations to 0.1% precision, which improves further to 0.02% in the pure LCDM scenario. Using our decomposition, we recast the one-loop redshift space galaxy power spectrum in a separable minimal-basis form, enabling $\sim 4000$ model evaluations per second at 0.02% precision on a single thread. This constitutes a considerable improvement over previously existing methods (e.g., FFTLog) opening a new window for efficient computations of higher loop and higher order correlators involving multiple powers of the linear matter power spectra. The resulting factorisation can also be utilised in clustering, weak lensing and CMB analyses. Our implementation is publicly available at https://github.com/ThomasBakx/cobra.