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Investigation of millimeter (mmWave) and Terahertz (THz) channels relies on channel measurements and estimation of multi-path component (MPC) parameters. As a common measurement technique in the mmWave and THz bands, direction-scan sounding (DSS) resolves angular information and increases the measurable distance. Through mechanical rotation, the DSS creates a virtual multi-antenna sounding system, which however incurs signal phase instability and large data sizes, which are not fully considered in existing estimation algorithms and thus make them ineffective. To tackle this research gap, in this paper, a DSS-oriented space-alternating generalized expectation-maximization (DSS-o-SAGE) algorithm is proposed for channel parameter estimation in mmWave and THz bands. To appropriately capture the measured data in mmWave and THz DSS, the phase instability is modeled by the scanning-direction-dependent signal phases. Furthermore, based on the signal model, the DSS-o-SAGE algorithm is developed, which not only addresses the problems brought by phase instability, but also achieves ultra-low computational complexity by exploiting the narrow antenna beam property of DSS. Simulations in synthetic channels are conducted to demonstrate the efficacy of the proposed algorithm and explore the applicable region of the far-field approximation in DSS-o-SAGE. Last but not least, the proposed DSS-o-SAGE algorithm is applied in real measurements in an indoor corridor scenario at 300~GHz. Compared with results using the baseline noise-elimination method, the channel is characterized more correctly and reasonably based on the DSS-o-SAGE.