Subgraph recognition aims at discovering a compressed substructure of a graph that is most informative to the graph property. It can be formulated by optimizing Graph Information Bottleneck (GIB) with a mutual information estimator. However, GIB suffers from training instability and de generated results due to its intrinsic optimization process. To tackle these issues, we reformulate the subgraph recognition problem into two steps: graph perturbation and sub graph selection, leading to a novel Variational Graph Information Bottleneck (VGIB) framework. VGIB first employs the noise injection to modulate the information flow from the input graph to the perturbed graph. Then, the perturbed graph is encouraged to be informative to the graph property. VGIB further obtains the desired subgraph by filtering out the noise in the perturbed graph. With the customized noise prior for each input, the VGIB objective is endowed with a tractable variational upper bound, leading to a superior empirical performance as well as theoretical properties. Extensive experiments on graph interpretation, explainability of Graph Neural Networks, and graph classification show that VGIB finds better subgraphs than existing methods.