The redshifts of all cosmologically distant sources are expected to experience a small, systematic drift as a function of time due to the evolution of the Universe's expansion rate. A measurement of this effect would represent a direct and entirely model-independent determination of the expansion history of the Universe over a redshift range that is inaccessible to other methods. Here we investigate the impact of the next generation of Extremely Large Telescopes on the feasibility of detecting and characterizing the cosmological redshift drift. We consider the Lyα forest in the redshift range 2 < z < 5 and other absorption lines in the spectra of high-redshift QSOs as the most suitable targets for a redshift drift experiment. Assuming photon-noise-limited observations and using extensive Monte Carlo simulations we determine the accuracy to which the redshift drift can be measured from the Lyα forest as a function of signal-to-noise ratio and redshift. Based on this relation and using the brightness and redshift distributions of known QSOs we find that a 42-m telescope is capable of unambiguously detecting the redshift drift over a period of ∼20 yr using 4000 h of observing time. Such an experiment would provide independent evidence for the existence of dark energy without assuming spatial flatness, using any other cosmological constraints or making any other astrophysical assumption.