We present the first-ever cosmological constraints from a simulation-based inference (SBI) analysis of galaxy clustering from the new SIMBIG forward modeling framework. SIMBIG leverages the predictive power of high-fidelity simulations and provides an inference framework that can extract cosmological information on small non-linear scales, inaccessible to standard analyses. In this work, we apply SIMBIG to the BOSS CMASS galaxy sample and analyze the power spectrum multipoles up to kmax=0.5h/Mpc. We construct 20,000 simulated galaxy samples using our forward model, which is based on high-resolution QUIJOTE-body simulations and includes detailed survey realism for a more complete treatment of observational systematics. We then conduct SBI by training normalizing flows using the simulated samples and infer the posterior distribution of LCDM cosmological parameters: Omega_m, Omega_b, h, n_s, sigma8. We derive significant constraints on Omega_m and sigma8, which are consistent with previous works. Our constraints on sigma8 are 27% more precise than standard analyses. This improvement is equivalent to the statistical gain expected from analyzing a galaxy sample that is nearly 60% larger than CMASS with standard methods. It results from additional cosmological information on non-linear scales beyond the limit of current analytic models, k > 0.25 h/Mpc. While we focus on power spectrum multipoles in this work for validation and comparison to the literature, SIMBIG provides a framework for analyzing galaxy clustering using any summary statistic. We expect further improvements on cosmological constraints from subsequent SIMBIG analyses of summary statistics beyond power spectrum.