In recent years, wave energy harvesting systems have received considerable attention as an alternative energy source. Within this class of systems, single-point harvesters are popular at least for preliminary studies and proof-of-concept analyses in particular locations. Unfortunately, the large displacements of a single-point wave energy harvester are described by a set of nonlinear equations. Further, the excitation is often characterized statistically and in terms of a relevant power spectral density (PSD) function. In the context of this complex problem, the development of efficient techniques for the calculation of reliable harvester response statistics is quite desirable, since traditional Monte Carlo techniques involve nontrivial computational cost. The paper proposes a statistical linearization technique for conducting expeditiously random vibration analyses of single-point harvesters. The technique is developed by relying on the determination of a surrogate linear system identified by minimizing the mean square error between the linear system and the nonlinear one. It is shown that the technique can be implemented via an iterative procedure, which allows calculating statistics, PSDs, and probability density functions (PDFs) of the response components. The reliability of the statistical linearization solution is assessed vis-à-vis data from relevant Monte Carlo simulations. This novel approach can be a basis for constructing computationally expeditious assessments of various design alternatives.