Context. The Gaia-ESO Public Spectroscopic Survey is obtaining high-quality spectroscopy of some 100 000 Milky Way stars using the FLAMES spectrograph at the VLT, down to V = 19 mag, systematically covering all the main components of the Milky Way and providing the first homogeneous overview of the distributions of kinematics and chemical element abundances in the Galaxy. Observations of young open clusters, in particular, are giving new insights into their initial structure, kinematics, and their subsequent evolution. Aims: This paper describes the analysis of UVES and GIRAFFE spectra acquired in the fields of young clusters whose population includes pre-main sequence (PMS) stars. The analysis is applied to all stars in such fields, regardless of any prior information on membership, and provides fundamental stellar atmospheric parameters, elemental abundances, and PMS-specific parameters such as veiling, accretion, and chromospheric activity. Methods: When feasible, different methods were used to derive raw parameters (e.g. line equivalent widths) fundamental atmospheric parameters and derived parameters (e.g. abundances). To derive some of these parameters, we used methods that have been extensively used in the past and new ones developed in the context of the Gaia-ESO survey enterprise. The internal precision of these quantities was estimated by inter-comparing the results obtained by these different methods, while the accuracy was estimated by comparison with independent external data, such as effective temperature and surface gravity derived from angular diameter measurements, on a sample of benchmarks stars. A validation procedure based on these comparisons was applied to discard spurious or doubtful results and produce recommended parameters. Specific strategies were implemented to resolve problems of fast rotation, accretion signatures, chromospheric activity, and veiling. Results: The analysis carried out on spectra acquired in young cluster fields during the first 18 months of observations, up to June 2013, is presented in preparation of the first release of advanced data products. These include targets in the fields of the ρ Oph, Cha I, NGC 2264, γ Vel, and NGC 2547 clusters. Stellar parameters obtained with the higher resolution and larger wavelength coverage from UVES are reproduced with comparable accuracy and precision using the smaller wavelength range and lower resolution of the GIRAFFE setup adopted for young stars, which allows us to provide stellar parameters with confidence for the much larger GIRAFFE sample. Precisions are estimated to be ≈120 K rms in Teff, ≈0.3 dex rms in log g, and ≈0.15 dex rms in [Fe/H] for the UVES and GIRAFFE setups.