MeV dark matter (DM) particles annihilating or decaying to electron-positron pairs cannot, in principle, be observed via local cosmic-ray (CR) measurements because of the shielding solar magnetic field. In this Letter, we take advantage of spacecraft Voyager 1's capacity for detecting interstellar CRs since it crossed the heliopause in 2012. This opens up a new avenue to probe DM in the sub-GeV energy/mass range that we exploit here for the first time. From a complete description of the transport of electrons and positrons at low energy, we derive predictions for both the secondary astrophysical background and the pair production mechanisms relevant to DM annihilation or decay down to the MeV mass range. Interestingly, we show that reacceleration may push positrons up to energies larger than the DM particle mass. We combine the constraints from the Voyager and AMS-02 data to get novel limits covering a very extended DM particle mass range, from MeV to TeV. In the MeV mass range, our limits reach annihilation cross sections of order ⟨σv⟩∼10^{-28} cm^{3}/s. An interesting aspect is that these limits barely depend on the details of cosmic-ray propagation in the weak reacceleration case, a configuration which seems to be favored by the most recent B/C data. Though extracted from a completely different and new probe, these bounds have a strength similar to those obtained with the cosmic microwave background-they are even more stringent for p-wave annihilation.