The physiochemical determinants of drug-target interactions in the microenvironment of the cell are complex and generally not defined by simple diffusion and intrinsic chemical reactivity. Non-specific interactions of drugs and macromolecules in cells are rarely considered formally in assessing pharmacodynamics. Here, we demonstrate that non-specific interactions lead to very slow incorporation kinetics of DNA binding drugs. We observe a rate of drug incorporation in cell nuclei three orders of magnitude slower than in vitro due to anomalous drug diffusion within cells. This slow diffusion, however, has an advantageous consequence: it leads to virtually irreversible binding of the drug to specific DNA targets in cells. We show that non-specific interactions drive slow drug diffusion manifesting as slow reaction front propagation. We study the effect of non-specific interactions in different cellular compartments by permeabilization of plasma and nuclear membranes in order to pinpoint differential compartment effects on variability in intracellular drug kinetics. These results provide the basis for a comprehensive model of the determinants of intracellular diffusion of small-molecule drugs, their target-seeking trajectories, and the consequences of these processes on the apparent kinetics of drug-target interactions.