We suggest in this paper a new stochastic simulation algorithm for solving narrow escape problems governed by drift-diffusion-reaction equations of high dimension. The developed method drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift directed to the target position. The method is especially appropriate to solve narrow escape problems for domains of very long extension in one direction which is the case in many practical problems. We present simulation results for a diffusion transport problem of calculation of cathodoluminescence intensity, a diffusion flux of excitons to a threading dislocation, and the electron beam induced current in a semiconductor. The diffusion tracking algorithm is based on the random walk on spheres process. The method is meshless both in space and time, and is well applied to solve high-dimensional problems in complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain.