Aims: We report 20 new lithium-rich giants discovered within the Gaia-ESO Survey, including the first Li-rich giant with an evolutionary stage confirmed by CoRoT (Convection, Rotation and planetary Transits) data. We present a detailed overview of the properties of these 20 stars. Methods: Atmospheric parameters and abundances were derived in model atmosphere analyses using medium-resolution GIRAFFE or high-resolution UVES (Ultraviolet and Visual Echelle Spectrograph) spectra. These results are part of the fifth internal data release of the Gaia-ESO Survey. The Li abundances were corrected for non-local thermodynamical equilibrium effects. Other stellar properties were investigated for additional peculiarities (the core of strong lines for signs of magnetic activity, infrared magnitudes, rotational velocities, chemical abundances, and Galactic velocities). We used Gaia DR2 parallaxes to estimate distances and luminosities. Results: The giants have A(Li) > 2.2 dex. The majority of them (14 of 20 stars) are in the CoRoT fields. Four giants are located in the field of three open clusters, but are not members. Two giants were observed in fields towards the Galactic bulge, but likely lie in the inner disc. One of the bulge field giants is super Li-rich with A(Li) = 4.0 dex. Conclusions: We identified one giant with infrared excess at 22 μm. Two other giants, with large v sin i, might be Li-rich because of planet engulfment. Another giant is found to be barium enhanced and thus could have accreted material from a former asymptotic giant branch companion. Otherwise, in addition to the Li enrichment, the evolutionary stages are the only other connection between these new Li-rich giants. The CoRoT data confirm that one Li-rich giant is at the core-He burning stage. The other giants are concentrated in close proximity to the red giant branch luminosity bump, the core-He burning stages, or the early-asymptotic giant branch. This is very clear from the Gaia-based luminosities of the Li-rich giants. This is also seen when the CoRoT Li-rich giants are compared to a larger sample of 2252 giants observed in the CoRoT fields by the Gaia-ESO Survey, which are distributed throughout the red giant branch in the Teff-log g diagram. These observations show that the evolutionary stage is a major factor for the Li enrichment in giants. Other processes, such as planet accretion, contribute at a smaller scale.