We derive an expression for the collision-induced amplitude dynamics in a fast collision between two (N + 1) −dimensional spatiotemporal solitons in saturable nonlinear media with weak perturbations in a spatial dimension of N, where N ≥ 1. The perturbed spatiotemporal soliton evolution is under a framework of the coupled saturable (N + 1 + 1) −dimensional nonlinear Schrödinger equations in the presence of weakly nonlinear loss and delayed Raman response. The perturbation approach is based on an extended perturbation technique for analyzing the collision-induced dynamics of one-dimensional temporal solitons and two-dimensional solitons. The accuracy of our theoretical calculations is validated by numerical simulations of the interaction of two 3D spatiotemporal solitons, also known as two light bullets, of the coupled nonlinear Schrödinger equations in the presence of delayed Raman response and cubic loss.