Quantitative phase imaging (QPI) has rapidly emerged as a complementary tool to fluorescence imaging, as it provides an objective measure of cell morphology and dynamics, free of variability due to contrast agents. In particular, three-dimensional (3D) tomographic imaging of live cells has opened up new directions of investigation by providing systematic and correlative analysis of various cellular parameters without limitations of photobleaching and phototoxicity. While current QPI systems allow the rapid acquisition of tomographic images, the pipeline to analyze these raw 3D tomograms is not well-developed. This work focuses on a critical, yet often underappreciated, step of the analysis pipeline, that of 3D cell segmentation from the acquired tomograms. The current method employed for such tasks is the Otsu-based 3D watershed algorithm, which works well for isolated cells; however, it is very challenging to draw boundaries when the cells are clumped. This process is also memory intensive since the processing requires computation on a 3D stack of images. We report the CellSNAP (Cell Segmentation via Novel Algorithm for Phase Imaging) algorithm for the segmentation of QPI images, which outstrips the current gold standard in terms of speed, robustness, and implementation, achieving cell segmentation under 2 seconds per cell on a single-core processor. The implementation of CellSNAP can easily be parallelized on a multi-core system for further speed improvements. For the cases where segmentation is possible with the existing standard method, our algorithm displays an average difference of 5% for dry mass and 8% for volume measurements. We also show that CellSNAP can handle challenging image datasets where cells are clumped and marred by interferogram drifts, which pose major difficulties for all QPI-focused segmentation tools. We envision our work will lead to the broader adoption of QPI imaging for high-throughput analysis, which has, in part, been stymied by a lack of suitable image segmentation tools.