Quantification of tissue morphology and biomarker distribution by means of optoacoustic tomography is an important and longstanding challenge, mainly caused by the complex heterogeneous structure of biological tissues as well as the lack of accurate and robust reconstruction algorithms. The recently introduced model-based inversion approaches were shown to mitigate some of reconstruction artifacts associated with the commonly used back-projection schemes, while providing an excellent platform for obtaining quantified maps of optical energy deposition in experimental configurations of various complexity. In this work, we introduce a weighted model-based approach, capable of overcoming reconstruction challenges caused by per-projection variations of object's illumination and other partial illumination effects. The universal weighting procedure is equally shown to reduce reconstruction artifacts associated with other experimental imperfections, such as non-uniform transducer sensitivity fields. Significant improvements in image fidelity and quantification are showcased both numerically and experimentally on tissue phantoms and mice.