We extend our recently proposed formalism for calculating anomalies of global and gauge symmetries using the covariant derivative expansion to include a general class of operators that can appear in relativistic effective field theories (EFTs). This allows us to prove that EFT operators involving general scalar, vector, and tensor couplings to fermion bilinears only give rise to irrelevant anomalies, which can be removed by an appropriate choice of counterterms, thereby confirming the absence of new constraints from anomaly cancellation on the Standard Model EFT.