We derive the Cherenkov Telescope Array (CTA) sensitivity to dark matter (DM) annihilation in several primary channels, over a broad range of DM masses. These sensitivities are estimated when CTA is pointed towards a large sample of Milky Ways dwarf spheroidal galaxies (dSphs) with promising $J$-factors and small statistical uncertainties. This analysis neglects systematic uncertainties, which we estimate at the level of at least 1 dex. We also present sensitivities on the annihilation cross section from a combined analysis of 4 dSphs. We assess the CTA sensitivity by: $i)$ using, for each dSph, a recent determination of the $J$-factor and its statistical error; $ii)$ considering the most up-to-date cosmic ray background; and $iii)$ including both spatial and spectral terms in the likelihood analysis. We find that a joint spectral and spatial analysis improves the CTA sensitivity, in particular for primary channels with sharp features in the $gamma$-ray energy spectrum and for dSphs with steep $J$-factor profiles, as deduced from the internal kinematics. The greatest sensitivities are obtained for observations of Ursa Minor among the classical dSphs and of Ursa Major II for ultra-faint dSphs.
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