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In this paper, a generalized logarithmic hyperbolic secant (GLHS) function is introduced that can effectively suppress impulsive noise while guarding the signal of interest from damage. Also, an analysis of the optimal scaling parameter choices for the GLHS function was studied. Then, in order to address the performance drawbacks of the traditional time delay estimation methods based on correlation under an impulsive noise environment, a novel GLHS-based correlation (GLHSC) is further developed, and the reliable time delay estimation result is obtained by finding the peak of GLHSC. The comprehensive Monte Carlo simulation results demonstrate that the performance of the method based on GLHSC is better than other robust competitive methods based on correlation in terms of probability of resolution and estimation accuracy, especially in a heavy-tailed noise environment.