Experimental studies with cold atoms have advanced our understanding of three-body physics, historically a fundamental yet challenging problem. This is because atomic interactions can be precisely varied in strength using magnetically tunable scattering resonances known as Feshbach resonances. Collisions near the unitarity limit, where scattering is maximum, are known to have universal aspects that are independent of short-range chemical details. Away from this limit, many quantum states are expected to be active during a three-body collision, making the collisional observables practically unpredictable. Here we predict three-body ultracold scattering rates by properly building in the pairwise van der Waals interactions plus the multi-spin properties of a tunable Feshbach resonance state characterized by known dimensionless two-body parameters. Numerically solving the Schrodinger equation then quantitatively determines three-atom collisional properties at all interaction strengths without needing adjustable parameters to fit data. Consequently, we can define a new class of van der Waals universality for cold atom three-body phenomena.