This paper examines the oscillatory behavior of solutions to Emden–Fowler neutral differential equations that incorporate a delayed damping term. To our knowledge, there are no prior results addressing the impact of delayed damping on the oscillatory behavior of neutral equations. We establish a new sufficient criterion to confirm that non-oscillatory solutions exhibit asymptotic behavior. Then, we refine and extend this criterion to include the ordinary case. Applying our results to an Euler-type equation shows that they are an extension and complement to the results reported in the literature.