Water management is a significant aspect of sustainable vegetable farming, especially in water-scarce regions. This, in addition to weed infestations, limits vegetable yields, which negatively affect food security in developing regions, particularly East Africa, where livelihoods chiefly depend on rain-fed agriculture. Vegetable farming, especially tomato cultivation, requires more water. By promoting mulching, a soil water conservation tool, we can control surface evaporation (E), which, together with irrigation, enhances effective water use and vegetable yields. The experiments for this study were conducted at the Tokyo University of Agriculture, Japan, to evaluate the influences of different irrigation conditions and poly-mulching on weed control, tomato yields, and water use efficiency. The study was conducted from May to September 2018 on a 30 m2 plot in an open-ended greenhouse using drip irrigation for tomato cultivation. Three predetermined irrigation conditions of 4, 3, and 2 mm/day were applied on black poly-mulched and bare ridges. Data on soil conditions—soil temperature, as well as meteorological variables, including solar radiation and temperature—were measured using thermocouple sensors and micro-hobo weather stations, respectively, during the tomato cultivation, while yield components—growth, yield, water productivity, and sugar content—were determined after harvest. The results of a two-way ANOVA show that irrigation conditions with poly-mulching reduced the weed biomass significantly, and improved yields and water use efficiency compared to the irrigation conditions on bare ridges. The application of 4, 3, and 2 mm/day irrigation with poly-mulching significantly reduced the weed biomass by 5% compared to the same irrigation conditions on bare ridges. Similarly, 4 and 3 mm/day irrigation conditions with poly-mulching significantly increased the tomato yield by 5% compared to 2 mm/day on bare ridges. The bigger roots were concentrated and widely distributed at the shallow soil depth (0–20 cm) of the ridges with high irrigation amounts, while the small and thin roots were in deeper soil layers (30–45 cm). This study provides scientific knowledge on the application of predetermined irrigation conditions that can be (i) integrated into irrigation scheduling and (ii) adopted for regions facing water scarcity and limited or no in situ meteorological data, to improve water use efficiency for vegetable cultivation.