The aim of this study was to develop and mathematically substantiate a method for assessing the randomness quality of output sequences produced by random and pseudorandom number generators based on spectral entropy. To solve the individual subproblems, the following methods were used: the discrete Fourier transform for signal spectrum computation; nonparametric bootstrap for constructing empirical distributions of estimates and determining decision thresholds; and the Mahalanobis distance for quantitative comparison of the analyzed realizations with a reference region and for formalizing the criterion of membership in the class of random sequences. Using a test case based on a deterministic sequence of natural numbers, it was established that Shannon entropy may yield a false-positive conclusion of randomness, whereas spectral entropy reliably detects non-randomness. The study of random sequences generated by a physical generator under different operating modes made it possible to conclude that the spectral entropy method detects degradation of sequence randomness when the generator operating conditions are chosen incorrectly. It should be noted that, in terms of effectiveness for the problem under consideration, the proposed spectral entropy method is not inferior to the standard NIST STS tests and, in some cases, outperforms them. Within the conducted experiments, it was established that the synthesized sequences generated using the ChaCha20 stream cipher satisfy the randomness criterion obtained by the spectral entropy method, which is interpreted as compliance with the requirements imposed on cryptographic sources of randomness. The study also showed that the developed method makes it possible to detect non-randomness in sequences generated by linear feedback shift registers (LFSRs). The scientific novelty of the work lies in the fact that а method for evaluating the quality of random and pseudorandom number generators based on spectral entropy computation has been theoretically substantiated and developed to a level of readiness for practical application. It has been established that, compared with Shannon entropy, the method provides greater robustness in randomness quality assessment. The practical significance lies in improving the efficiency of assessing the quality of random and pseudorandom sequences.