Context. Open clusters are recognised as excellent tracers of Galactic thin-disc properties. At variance with intermediate-age and old open clusters, for which a significant number of studies is now available, clusters younger than ≲150 Myr have been mostly overlooked in terms of their chemical composition until recently (with few exceptions). On the other hand, previous investigations seem to indicate an anomalous behaviour of young clusters, which includes (but is not limited to) slightly sub-solar iron (Fe) abundances and extreme, unexpectedly high barium (Ba) enhancements. Aims. In a series of papers, we plan to expand our understanding of this topic and investigate whether these chemical peculiarities are instead related to abundance analysis techniques. Methods. We present a new determination of the atmospheric parameters for 23 dwarf stars observed by the Gaia-ESO survey in five young open clusters (τ < 150 Myr) and one star-forming region (NGC 2264). We exploit a new method based on titanium (Ti) lines to derive the spectroscopic surface gravity, and most importantly, the microturbulence parameter. A combination of Ti and Fe lines is used to obtain effective temperatures. We also infer the abundances of Fe I, Fe II, Ti I, Ti II, Na I, Mg I, Al I, Si I, Ca I, Cr I, and Ni I. Results. Our findings are in fair agreement with Gaia-ESO iDR5 results for effective temperatures and surface gravities, but suggest that for very young stars, the microturbulence parameter is over-estimated when Fe lines are employed. This affects the derived chemical composition and causes the metal content of very young clusters to be under-estimated. Conclusions. Our clusters display a metallicity [Fe/H] between +0.04 ± 0.01 and +0.12 ± 0.02; they are not more metal poor than the Sun. Although based on a relatively small sample size, our explorative study suggests that we may not need to call for ad hoc explanations to reconcile the chemical composition of young open clusters with Galactic chemical evolution models.