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The material processing technique of “stealth” nanomachining is based on translating a longitudinally extended beam such as a Bessel beam into a transparent sample to generate extended nanochannels, which leads to subsequent internal stress that facilitates high quality cleaving. In this letter, we compare the quality of such cleaving in glass samples obtained using Bessel beams with both circularly symmetric and elliptical transverse profiles. We find that the use of an elliptical Bessel beam generates elliptical nanochannels, which greatly improves the cleavage quality and cuts material strength by aligning the centre of the cleavage plane with the centre of the machined channels. These results are interpreted using numerical simulations that show how elliptical nanochannels enhance the intensity and localization of the tensile stress distribution in glass under bending when compared to channels with circular cross-sections.