The infrared Calcium Triplet and its nearby spectral region have been used for spectral and luminosity classification of late-type stars, but the samples of cool supergiants (CSGs) used have been very limited (in size, metallicity range, and spectral types covered). The spectral range of the Gaia Radial Velocity Spectrograph (RVS) covers most of this region but does not reach the main TiO bands in this region, whose depths define the M sequence. We study the behaviour of spectral features around the Calcium Triplet and develop effective criteria to identify and classify CSGs, comparing their efficiency with other methods previously proposed. We measure the main spectral features in a large sample (almost 600) of CSGs from three different galaxies, and we analyse their behaviour through a principal component analysis. Using the principal components, we develop an automatised method to differentiate CSGs from other bright late-type stars, and to classify them. The proposed method identifies a high fraction of the supergiants (SGs) in our test sample, which cover a wide metallicity range (SGs from the SMC, the LMC, and the Milky Way) and with spectral types from G0 up to late-M. In addition, it is capable to separate most of the non-SGs in the sample, identifying as SGs only a very small fraction of them. A comparison of this method with other previously proposed shows that it is more efficient and selects less interlopers. A way to automatically assign a spectral type to the SGs is also developed. We apply this study to spectra at the resolution and spectral range of the Gaia RVS, with a similar success rate. The method developed identifies and classifies CSGs in large samples, with high efficiency and low contamination, even in conditions of wide metallicity and spectral-type ranges.