  Minimal gauged U(1)$_{L_{\mu}-L_{\tau}}$ models can provide for an additional
source for the muon anomalous magnetic moment however it is difficult to
accommodate the discrepancy in the electron magnetic moment in tandem. Owing to
the relative sign between the discrepancies in these quantities, it seems
unlikely that they arise from the same source. We show that a supersymmetric
(SUSY) gauged U(1)$_{L_{\mu}-L_{\tau}}$ model can accommodate both the muon and
electron anomalous magnetic moments in a very simple and intuitive scenario,
without utilizing lepton flavor violation. The currently allowed parameter
space in this kind of a scenario is constrained from the latest LHC and various
low energy experimental data,e.g., recent COHERENT data, CCFR, Borexino, BaBaR,
supernova etc. These constraints, in conjunction with the requirement to
explain both lepton magnetic moments, lead to an upper bound on the first
generation slepton mass, a lower bound on the second generation slepton mass
and constricts the allowed range for the new gauge boson mass and coupling. The
scheme can be probed at the ongoing COHERENT and Coherent CAPTAIN-Mills
experiments and at future experiments, e.g., DUNE, BELLE-II etc.
