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In this paper we introduce the SEAGLE (i.e. Simulating EAGLE LEnses) programme, which approaches the study of galaxy formation through strong gravitational lensing, using a suite of high-resolution hydrodynamic simulations, Evolution and Assembly of GaLaxies and their Environments (EAGLE) project. We introduce the simulation and analysis pipeline and present the first set of results from our analysis of early-type galaxies. We identify and extract an ensemble of simulated lens galaxies and use the glamer ray-tracing lensing code to create mock lenses similar to those observed in the Sloan Lens ACS Survey (SLACS) and SL2S surveys, using a range of source parameters and galaxy orientations, including observational effects such as the point spread function, pixelization, and noise levels, representative of single-orbit observations with the Hubble Space Telescope (HST) using the ACS- F 814 W filter. We subsequently model these mock lenses using the code lensed, treating them in the same way as observed lenses. We also estimate the mass model parameters directly from the projected surface mass density of the simulated galaxy, using an identical mass model family. We perform a three-way comparison of all the measured quantities with real lenses. We find the average total density slope of EAGLE lenses, $$t=2.26\,\, (0.25\,\, {\rm rms})$$ to be higher than SL2S, t  = 2.16 or SLACS, t  = 2.08. We find a very strong correlation between the external shear (γ) and the complex ellipticity (ε), with γ ∼ ε/4. This correlation indicates a degeneracy in the lens mass modelling. We also see a dispersion between lens modelling and direct fitting results, indicating systematical biases. [ABSTRACT FROM AUTHOR]