Direction-of-arrival (DOA) estimation performance of adaptive filtering-based methods, e.g., least mean square (LMS), degrades significantly in adverse conditions (i.e., at low signal-to-noise ratio, small number of antenna array elements and snapshots, and in presence of two closely spaced signal sources). Moreover, these estimation methods require many regulating parameters to efficiently update the step size, which are very difficult to tune manually in practical scenarios. Although the subspace decomposition-based methods, e.g., multiple signal classification, provide somewhat better performance in adverse conditions, they are based on the statistical properties of the signal, such as spatial covariance matrix (SCM) and its eigenvalue decomposition, resulting in high computational complexity. In this article, a robust DOA estimation method is proposed based on a variable step size LMS algorithm with low-rank matrix approximation (LRMA) where the received signal denoising problem is formulated first as an LRMA problem directly using the received signal observations instead of the SCM, and then the variable step size is calculated from the predicted instantaneous error and the estimated powers of reference and denoised signals. The output spatial spectrum is determined by the reciprocal of the antenna array pattern, and the high peaks in the output spatial spectrum give the received signals estimated directions. The proposed method updates the step size efficiently without choosing any regulating parameter and achieves improved performance even in adverse conditions due to the denoising feature of LRMA with low computational complexity. Furthermore, we also conduct the convergence and stability analyses of the proposed iterative estimation method with the denoised signal. Numerical results demonstrate that the proposed method outperforms state-of-the-art methods, especially those which yielded based on adaptive filtering.