MOTIVATION: In recent years, network models have gained prominence for their ability to capture complex associations. In statistical omics, networks can be used to model and study the functional relationships between genes, proteins, and other types of omics data. If a Gaussian graphical model is assumed, a gene association network can be determined from the non-zero entries of the inverse covariance matrix of the data. Due to the high-dimensional nature of such problems, integrative methods that leverage similarities between multiple graphical structures have become increasingly popular. The joint graphical lasso is a powerful tool for this purpose, however, the current AIC-based selection criterion used to tune the network sparsities and similarities leads to poor performance in high-dimensional settings. RESULTS: We propose stabJGL, which equips the joint graphical lasso with a stable and well-performing penalty parameter selection approach that combines the notion of model stability with likelihood-based similarity selection. The resulting method makes the powerful joint graphical lasso available for use in omics settings, and outperforms the standard joint graphical lasso, as well as state-of-the-art joint methods, in terms of all performance measures we consider. Applying stabJGL to proteomic data from a pan-cancer study, we demonstrate the potential for novel discoveries the method brings. AVAILABILITY AND IMPLEMENTATION: A user-friendly R package for stabJGL with tutorials is available on Github https://github.com/Camiling/stabJGL.