The effect of an external force on the kinetics of diffusion-assisted escaping of Brownian particles from a potential well is analyzed in detail. The analysis is made within the two-state model (TSM) of the process which is known to be valid in the deep well limit in the absence of external force. The generalized variant of the TSM, taking into account the effect of the force, is shown to be quite accurate as well for some shapes of the well. Within the generalized TSM, simple expressions for the well depopulation kinetics and, in particular, for the escape rate ire obtained. These expressions show that the effect of a force (F) manifests itself in the escape rate dependence on only one parameter phi = Fa/(2k(b)T), where a is the Onsager radius of the attractive part of the well U(r), defined by the relation vertical bar U(a)vertical bar approximate to k(b)T. The limiting behavior of this dependence in the cases of weak and strong force is analyzed in detail. Possible applications as well as the relation of the results of the analysis to those obtained earlier are briefly discussed.