To achieve the ambitious aims of the Paris climate agreement, the majority of fossil-fuel reserves needs to remain underground. As current national government commitments to mitigate greenhouse gas emissions are insufficient by far, actors such as institutional and private investors and the social movement on divestment from fossil fuels could play an important role in putting pressure on national governments on the road to decarbonization. Using a stochastic agent-based model of co-evolving financial market and investors' beliefs about future climate policy on an adaptive social network, here we find that the dynamics of divestment from fossil fuels shows potential for social tipping away from a fossil-fuel based economy. Our results further suggest that socially responsible investors have leverage: a small share of 10--20\,\% of such moral investors is sufficient to initiate the burst of the carbon bubble, consistent with the Pareto Principle. These findings demonstrate that divestment has potential for contributing to decarbonization alongside other social movements and policy instruments, particularly given the credible imminence of strong international climate policy. Our analysis also indicates the possible existence of a carbon bubble with potentially destabilizing effects to the economy.