Distinguishing a dark matter interaction from an astrophysical neutrino-induced interaction will be major challenge for future direct dark matter searches. In this paper, we consider this issue within non-relativistic Effective Field Theory (EFT), which provides a well-motivated theoretical framework for determining nuclear responses to dark matter scattering events. We analyze the nuclear energy recoil spectra from the different dark matter-nucleon EFT operators, and compare to the nuclear recoil energy spectra that is predicted to be induced by astrophysical neutrino sources. We determine that for 11 of the 14 possible operators, the dark matter-induced recoil spectra can be cleanly distinguished from the corresponding neutrino-induced recoil spectra with moderate size detector technologies that are now being pursued, e.g., these operators would require 0.5 tonne years to be distinguished from the neutrino background for low mass dark matter. Our results imply that in most models detectors with good energy resolution will be able to distinguish a dark matter signal from a neutrino signal, without the need for much larger detectors that must rely on additional information from timing or direction. In addition we calculate up-to-date exclusion limits in the EFT model space using data from the LUX experiment.