We study the effect of dimension-six operators of the standard model effective field theory relevant for vector-boson fusion and associated Higgs boson production at the LHC. We present predictions at the next-to-leading order accuracy in QCD that include matching to parton showers and that rely on fully automated simulations. We show the importance of the subsequent reduction of the theoretical uncertainties in improving the possible discrimination between effective field theory and standard model results and investigate the impact of including the squared contributions of the new operators given current constraints from global fits to low energy and LHC data. The difference between including and not including these formally higher order contributions was found to be significant in the associated production case, suggesting that an effective field theory interpretation of current measurements lies on the edge of its validity range.