Stellar evolution models predict that internal mixing should cause some sodium overabundance at the surface of red giants more massive than ~1.5-2.0M_{aun}_. The surface aluminium abundance should not be affected. Nevertheless, observational results disagree about the presence and/or the degree of Na and Al overabundances. In addition, Galactic chemical evolution models adopting different stellar yields lead to very different predictions for the behavior of [Na/Fe] and [Al/Fe] versus [Fe/H]. Overall, the observed trends of these abundances with metallicity are not well reproduced. We re-address both issues, using new Na and Al abundances determined within the Gaia-ESO Survey. Our aim is to obtain better observational constraints on the behavior of these elements using two samples: i) more than 600 dwarfs of the solar neighborhood and of open clusters and ii) low- and intermediate-mass clump giants in six open clusters. Abundances were determined using high-resolution UVES spectra. The individual Na abundances were corrected for nonlocal thermodynamic equilibrium effects. For the Al abundances, the order of magnitude of the corrections was estimated for a few representative cases. For giants, the abundance trends with stellar mass are compared to stellar evolution models. For dwarfs, the abundance trends with metallicity and age are compared to detailed chemical evolution models.