Recent observational studies suggest that feedback from active galactic nuclei (AGNs) may play an important role in the formation and evolution of low-mass dwarf galaxies, an issue that has received little attention from a theoretical perspective. We investigate this using two sets of 12 cosmological magneto-hydrodynamical simulations of the formation of dwarf galaxies: one set using a version of the AURIGA galaxy formation physics model including AGN feedback and a parallel set with AGN feedback turned off. We show that the full-physics AGN runs satisfactorily reproduce several scaling relations, including the M_ BH-M_*, M_ BH-sigma_* and the baryonic Tully-Fisher relation. We find that the global star formation (SF) of galaxies run with AGN is reduced compared to the one in which AGN has been turned off, suggesting that this type of feedback is a viable way of suppressing SF in low-mass dwarfs, even though none of our galaxies is completely quenched by z=0. Furthermore, we found a tight correlation between the median SF rates and the black-hole-to-stellar mass ratio (M_BH/M_star) in our simulated dwarfs. SF is suppressed due to gas heating in the vicinity of the AGN: less HI gas is available in AGN runs, though the total amount of gas is preserved across the two settings within each galaxy. This indicates that the main effect of AGN feedback in our dwarfs is to heat up and push the gas away from the galaxy's centre rather than expelling it completely. Finally, we show that the two galaxies harbouring the largest BHs have suffered a considerable (up to 65%) reduction in their central dark matter density, pinpointing the role of AGNs in determining the final dark matter mass distribution within dwarf galaxies. This pilot work highlights the importance of modelling AGN feedback at the lowest mass scales and the impact this can have on dwarf galaxy evolution.