Mediation analyses play important roles in making causal inference in biomedical research to examine causal pathways that may be mediated by one or more intermediate variables (i.e., mediators). Although mediation frameworks have been well established such as counterfactual-outcomes (i.e., potential-outcomes) models and traditional linear mediation models, little effort has been devoted to dealing with mediators with zero-inflated structures due to challenges associated with excessive zeros. We develop a novel mediation modeling approach to address zero-inflated mediators containing true zeros and false zeros. The new approach can decompose the total mediation effect into two components induced by zero-inflated structures: the first component is attributable to the change in the mediator on its numerical scale which is a sum of two causal pathways and the second component is attributable only to its binary change from zero to a non-zero status. An extensive simulation study is conducted to assess the performance and it shows that the proposed approach outperforms existing standard causal mediation analysis approaches. We also showcase the application of the proposed approach to a real study in comparison with a standard causal mediation analysis approach.