A speech enhancement (SE) technique improves the quality and intelligibility of the noise degraded speech. The conventional MMSE-based SE technique minimizes the Bayesian cost function by modifying the magnitude-spectra, whereas keeping the phase-spectra intact. In this current study, both magnitude and phase spectra are modified in order to enhance the noisy speech using a phase compensated perceptually weighted β-order Bayesian estimator. In the proposed SE technique, we first modified the phase of noisy speech spectra alone. Secondly, we used perceptually motivated β-order Bayesian estimator to modify the magnitude-spectra. The estimator holds the advantages of the perceptually-weighted and the β-order spectral amplitude estimator to get a better gain function. The compensated phase spectra and the estimated magnitude spectra are combined to reconstruct the noise attenuated enhanced speech signals. The proposed speech enhancement technique is evaluated for different noise levels (0 dB to +10 dB) in terms of the objective quality and intelligibility measures using the NOIZEUS and AURORA databases. It is observed that the proposed enhancement technique considerably improves the quality and maintains intelligibility in non-stationary and stationary noisy environments. The experimental results show that the average short-time objective intelligibility (STOI), normalized covariance metric (NCM), perceptual evaluation of speech quality (PESQ), segmental SNR (SegSNR), and Likelihood-Ratio (LLR) are improved by 12.2%, 9.25%, 19.69%, 31.25%, and 39.06% over the noisy unprocessed speech signals. The proposed technique also outscored the competing techniques.