We have studied the effects of various initial mass functions (IMFs) on the chemical evolution of the Sagittarius dwarf galaxy (Sgr). In particular, we tested the effects of the integrated galactic initial mass function (IGIMF) on various predicted abundance patterns. The IGIMF depends on the star formation rate and metallicity and predicts less massive stars in a regime of low star formation, as it is the case in dwarf spheroidals. We adopted a detailed chemical evolution model following the evolution of $α$-elements, Fe and Eu, and assuming the currently best set of stellar yields. We also explored different yield prescriptions for the Eu, including production from neutron star mergers. Although the uncertainties still present in the stellar yields and data prevent us from drawing firm conclusions, our results suggest that the IGIMF applied to Sgr predicts lower [$α$/Fe] ratios than classical IMFs and lower [hydrostatic/explosive] $α$-element ratios, in qualitative agreement with observations. In our model, the observed high [Eu/O] ratios in Sgr is due to reduced O production, resulting from the IGIMF mass cutoff of the massive oxygen-producing stars, as well as to the Eu yield produced in neutron star mergers, a more promising site than core-collapse supernovae, although many uncertainties are still present in the Eu nucleosynthesis. We find that a model, similar to our previous calculations, based on the late addition of iron from the Type Ia supernova time-delay (necessary to reproduce the shape of [X/Fe] versus [Fe/H] relations) but also including the reduction of massive stars due to the IGIMF, better reproduces the observed abundance ratios in Sgr than models without the IGIMF. ; Comment: Accepted for publication in MNRAS, 16 pages, 16 figures; typos fixed and references updated (v2); Fig. 16 revised (v3); version matching the one appearing on the electronic Journal