Co-Al-W-based alloys have been recognized as promising next-generation structural materials for high-temperature applications due to the formation of the γ′ strengthened phase. However, optimizing these superalloys through experimental studies is a significant challenge because of complex element interactions. This paper investigates the temporal processing and mechanisms of γ′ precipitates in novel Co-based superalloys using the phase-field method. By coupling the calculation of phase diagrams (CALPHAD) approach for the phase free energy and chemical mobility involving element interactions, and by calculating thermophysical parameters of alloys using first-principles, we developed a quantitative phase-field model for multicomponent systems. Using this model, the element diffusion path and γ′ evolution path are simulated in ternary diffusion couples and aging alloys, and the results align well with experimental observations. Furthermore, the effects of W content in Co-Al-W alloys on the coarsening kinetics and mechanisms of γ′ precipitates during long-term aging are systematically studied. It is found that increasing W content results in higher γ′ volume fraction and increased γ′ coarsening rate. The accelerated coarsening is primarily attributed to the shortened inter-particle spacing between γ′ precipitates, which is comprehensively analyzed through element diffusion distance and flux. Additionally, the model is extended to quaternary systems and successfully applied to Co-Ni-Al-W alloy. This study provides a novel method for the quantitative prediction of γ/γ′ microstructures and contributes to the alloy design and processing optimization of novel superalloys.