In this study, the cutting of borosilicate glass plates in ambient air and water with a 355 nm wavelength picosecond laser was carried out. Low (2.1–2.75 W) and high (15.5 W) average laser power cutting regimes were studied. Thorough attention was paid to the effect of the hatch distance on the cutting quality and characteristic strength of glass strips cut in both environments. At optimal cutting parameters, ablation efficiency and cutting rates were the highest but cut sidewalls were covered with periodically recurring ridges. Transition to smaller hatch values improved the cut sidewall quality by suppressing the ridge formation, but negatively affected the ablation efficiency and overall strength of glass strips. Glass strips cut in water in the low-laser-power regime had the highest characteristic strength of 117.6 and 107.3 MPa for the front and back sides, respectively. Cutting in a high-laser-power regime was only carried out in water. At 15.5 W, the ablation efficiency and effective cutting speed per incident laser power increased by 16% and 22%, respectively, compared with cutting in water in a low-laser-power regime.