Context. The Gaia-ESO Public Spectroscopic Survey using FLAMES at the VLT has obtained high-resolution UVES spectra for a large number of giant stars, allowing a determination of the abundances of the key chemical elements carbon and nitrogen at their surface. The surface abundances of these chemical species are known to change in stars during their evolution on the red giant branch (RGB) after the first dredge-up episode, as a result of the extra mixing phenomena. Aims: We investigate the effects of thermohaline mixing on C and N abundances using the first comparison between the Gaia-ESO survey [C/N] determinations with simulations of the observed fields using a model of stellar population synthesis. Methods: We explore the effects of thermohaline mixing on the chemical properties of giants through stellar evolutionary models computed with the stellar evolution code STAREVOL. We include these stellar evolution models in the Besançon Galaxy model to simulate the [C/N] distributions determined from the UVES spectra of the Gaia-ESO survey and to compare them with the observations. Results: Theoretical predictions including the effect of thermohaline mixing are in good agreement with the observations. However, the field stars in the Gaia-ESO survey with C and N abundance measurements have a metallicity close to solar, where the efficiency of thermohaline mixing is not very large. The C and N abundances derived by the Gaia-ESO survey in open and globular clusters clearly show the impact of thermohaline mixing at low metallicity, which explains the [C/N] value observed in lower mass and older giant stars. Using independent observations of carbon isotopic ratio in clump field stars and open clusters, we also confirm that thermohaline mixing should be taken into account to explain the behaviour of 12C/13C as a function of stellar age. Conclusions: Overall, the current model including thermohaline mixing is able to reproduce very well the C and N abundances over the whole metallicity range investigated by the Gaia-ESO survey data.