Baryonic feedback effects lead to a suppression of the weak lensing angular power spectrum on small scales. The poorly constrained shape and amplitude of this suppression is an important source of uncertainties for upcoming cosmological weak lensing surveys such as Euclid or LSST. In this first paper in a series of two, we use simulations to build a Euclid-like tomographic mock data-set for the cosmic shear power spectrum and the corresponding covariance matrix, which are both corrected for baryonic effects following the baryonification method of . In addition, we develop an emulator to obtain fast predictions of the baryonic power suppression, allowing us to perform a likelihood inference analysis for a standard ΛCDM cosmology with both cosmological and astrophysical parameters. Our main findings are the following: (i) ignoring baryonic effects leads to a greater than 5σ bias on the cosmological parameters Ωm and σ8; (ii) restricting the analysis to the largest scales, that are mostly unaffected by baryons, makes the bias disappear, but results in a blow-up of the Ωm-σ8 contour area by more than a factor of 10; (iii) ignoring baryonic effects on the covariance matrix does not significantly affect cosmological parameter estimates; (iv) while the baryonic suppression is mildly cosmology dependent, this effect does not noticeably modify the posterior contours. Overall, we conclude that including baryonic uncertainties in terms of nuisance parameters results in unbiased and surprisingly tight constraints on cosmology.