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The dynamical arrest of attractive colloidal particles into out-of-equilibrium structures, known as gelation, is central to biophysics, materials science, nanotechnology, and food and cosmetic applications, but a complete understanding is lacking. In particular, for intermediate particle density and attraction, the structure formation process remains unclear. Here, we show that the gelation of short-range attractive particles is governed by a nonequilibrium percolation process. We combine experiments on critical Casimir colloidal suspensions, numerical simulations, and analytical modeling with a master kinetic equation to show that cluster sizes and correlation lengths diverge with exponents 1.6 and 0.8, respectively, consistent with percolation theory, while detailed balance in the particle attachment and detachment processes is broken. Cluster masses exhibit power-law distributions with exponents -3/2 and -5/2 before and after percolation, as predicted by solutions to the master kinetic equation. These results revealing a nonequilibrium continuous phase transition unify the structural arrest and yielding into related frameworks.