Effective field theory (EFT) provides a model-independent framework for interpreting the results of dark matter (DM) direct detection experiments. In this study, we demonstrate that the two fermionic DM-quark tensor operators $(\bar{\chi} {\rm{i}}\sigma^{\mu\nu} \gamma^5 \chi) (\bar{q} \sigma_{\mu\nu}q)$ and $(\bar{\chi} {\rm{i}}\sigma^{\mu\nu} \gamma^5 \chi) (\bar{q} \sigma_{\mu\nu}q)$ can contribute to the DM electric and magnetic dipole moments via nonperturbative QCD effects, in addition to the well-studied contact DM-nucleon operators. We then investigate the constraints on these two operators by considering both the contact and dipole contributions using the XENON1T nuclear recoil and Migdal effect data. We also recast other existing bounds on the DM dipole operators, derived from electron and nuclear recoil measurements in various direct detection experiments, as constraints on the two tensor operators. For $(\bar{\chi} {\rm{i}}\sigma^{\mu\nu} \gamma^5 \chi) (\bar{q} \sigma_{\mu\nu}q)$ , our results significantly extend the reach of constraints on the DM-quark tensor operators to masses as low as $(\bar{\chi} {\rm{i}}\sigma^{\mu\nu} \gamma^5 \chi) (\bar{q} \sigma_{\mu\nu}q)$ , with the bound exceeding that obtained by the Migdal effect with only contact interactions by approximately an order of magnitude. In particular, for the operator $(\bar{\chi} {\rm{i}}\sigma^{\mu\nu} \gamma^5 \chi) (\bar{q} \sigma_{\mu\nu}q)$ with DM mass $(\bar{\chi} {\rm{i}}\sigma^{\mu\nu} \gamma^5 \chi) (\bar{q} \sigma_{\mu\nu}q)$ , the latest PandaX constraint on the DM electric dipole moment puts more stringent bounds than the previous direct detection limit. We also briefly discuss the constraints obtained from experiments other than direct detection.