Dwarf galaxies provide insight into the processes of star formation and chemical enrichment at the low end of the galaxy mass function, as well as into the clustering of dark matter on small scales. In studies of Local Group dwarf galaxies, spectroscopic samples of individual stars are used to derive the internal kinematics and abundance properties of these galaxies. It is therefore important to clean these samples from Milky Way stars, which are not related to the dwarf galaxy, since they can contaminate analysis of the properties of these objects. Here we introduce a new diagnostic for separating Milky Way contaminant stars, which mainly consist of dwarf stars, and red giant branch stars targeted in dwarf galaxies. As discriminator we use the trends in the equivalent width of the nIR Mg I line at 8806.8 Å as a function of the equivalent width of Ca II triplet lines. This method is particularly useful for works dealing with multi-object, intermediate-resolution spectroscopy focusing in the region of the nIR Ca II triplet. We use synthetic spectra to explore how the equivalent width of these lines changes for stars with different properties (gravity, effective temperature, metallicity) and find that a distinction among giants above the horizontal branch and dwarfs can be made with this method at [Fe/H] > -2 dex. For -2 ≤ [Fe/H] ≤ -1, this method is also valid for distinguishing dwarfs and giants down to approximately one magnitude below the horizontal branch. Using a foreground model we make predictions on the use of this new discrimination method for nearby dwarf spheroidal galaxies, including the ultra-faints. We subsequently use VLT/FLAMES data for the Sextans, Sculptor, and Fornax dwarf spheroidal galaxies to verify the predicted theoretical trends. Based on FLAMES observations collected at the ESO, proposals 171.B-0588, 076.B-0391, 079.B-0435.