We optically drive a microsphere at constant speed through entangled actin networks of 0.2-1.4 mg/mL at rates faster than the critical rate controlling the onset of nonlinear response. By measuring the resistive force exerted on the microsphere during and following strain, we reveal a critical concentration c(c) similar or equal to 0.4 mg/mL for nonlinear features to emerge. For c > c(c), entangled actin stiffens at short times with the degree of stiffening S and corresponding time scale t(stiff) scaling with the entanglement tube density, i.e., S similar to t(stiff) similar to d(t)(-1) similar to c(3/5). The network subsequently yields to a viscous regime with the yield distance d(y) scaling linearly with yield force f(y) and inversely with the entanglement length (f(y) similar to d(y) similar to l(e)(-1) similar to c(2/5)). Stiffening and yielding dynamics are consistent with recent theoretical predictions for nonlinear cohesive breakdown of entanglements. We further show that above c(c) force relaxation proceeds via slow filament disengagement from dilated tubes coupled with similar to 10x faster lateral hopping, with the corresponding concentration dependences in agreement with recent theoretical predictions for entangled rigid rods.