We report the discovery of two Li-rich giants, with A(Li) ~ 1.50, in an analysis of a sample of 40 giants of the open cluster Trumpler 20 (with turnoff mass ~ 1.8 Msun). The cluster was observed in the context of the Gaia-ESO Survey. The atmospheric parameters and Li abundances were derived using high-resolution UVES spectra. The Li abundances were corrected for nonlocal thermodynamical equilibrium (non-LTE) effects. Only upper limits of the Li abundance could be determined for the majority of the sample. Two giants with detected Li turned out to be Li rich: star MG 340 has A(Li) non-LTE = 1.54 \pm 0.21 dex and star MG 591 has A(Li) non-LTE = 1.60 \pm 0.21 dex. Star MG 340 is on average ~ 0.30 dex more rich in Li than stars of similar temperature, while for star MG 591 this difference is on average ~ 0.80 dex. Carbon and nitrogen abundances indicate that all stars in the sample have completed the first dredge-up. The Li abundances in this unique sample of 40 giants in one open cluster clearly show that extra mixing is the norm in this mass range. Giants with Li abundances in agreement with the predictions of standard models are the exception. To explain the two Li-rich giants, we suggest that all events of extra mixing have been inhibited. This includes rotation-induced mixing during the main sequence and the extra mixing at the red giant branch luminosity bump. Such inhibition has been suggested in the literature to occur because of fossil magnetic fields in red giants that are descendants of main-sequence Ap-type stars.