The concept of sparsity is extensively exploited in the fields of data acquisition and image processing, contributing to better signal-to-noise and spatio-temporal performance of the various imaging methods. In the field of optoacoustic tomography, the image reconstruction problem is often characterized by computationally extensive inversion of very large datasets, for instance when acquiring volumetric multispectral data with high temporal resolution. In this article we seek to accelerate accurate model-based optoacoustic inversions by identifying various sources of sparsity in the forward and inverse models as well as in the single- and multi-frame representation of the projection data. These sources of sparsity are revealed through appropriate transformations in the signal, model and image domains and are subsequently exploited for expediting image reconstruction. The sparsity-based inversion scheme was tested with experimental data, offering reconstruction speed enhancement by a factor of 40 to 700 times as compared with the conventional iterative model-based inversions while preserving similar image quality. The demonstrated results pave the way for achieving real-time performance of model-based reconstruction in multi-dimensional optoacoustic imaging.