In a broad class of models, direction selectivity in primary visual cortical neurons arises from the linear summation of spatially offset and temporally lagged inputs combined with a spike threshold. Here, we characterize the robustness of this class of models to input noise and background activity that is uncorrelated with the visual stimulus. When only excitatory inputs were considered, moderate levels of noise substantially degraded direction selectivity. By contrast, the inclusion of inhibition produced a direction-selective neuron even at high noise levels. Moreover, if inhibitory inputs were tuned, mirroring excitatory inputs but lagging by a fixed delay, they promoted a highly direction-selective response by suppressing all excitatory inputs in the null direction while minimally affecting excitatory inputs in the preferred direction. Additionally, tuned inhibition strongly reduced trial-by-trial variability, such that the neuron produced a consistent direction-selective response to multiple presentation of the same stimulus. This work illustrates how inhibition could be used by cortical circuits to reliably extract information on a single-trial basis from feed-forward inputs in a noisy, high-background context.